Nondestructive Evaluation of Composite Rods Using
Ultrasonic Wave Propagation
by
Vanea R. Pharr
B.S. Applied Mathematics
Old Dominion University, 2009
SUBMITTED TO THE DEPARTMENT OF MECHANICAL ENGINEERING IN PARTIAL FULFILLMENT OF
THE REQUIREMENTS FOR THE DEGREES OF
ARCHIVES
NAVAL ENGINEER
MASSACHUSETT
AND
NST TUTE
OF TECHNOLOLGY
MASTER OF SCIENCE IN MECHANICAL ENGINEERING
AT THE
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
JUL 3 0 2015
LIBRARIES
JUNE 2015
@ Vanea R. Pharr 2014. All rights reserved.
The author hereby grants to MIT permission to reproduce and to distribute publicly paper and
electronic copies of this thesis document in whole or in part in any medium now known or
hereafter created.
Signature redacted
Signature of Author:
N
Certified by:
If
IF
f
f
If
A
Dep artment of Mechanical Engineering
May 4, 2015
Signature redacted
UU
James H. Williams
Profe ssor of Applied Mechanics
Thesis Supervisor
Signature redacted
Accepted by:
David E. Hardt
Chairman, Committee on Graduate Students
Nondestructive Evaluation of Composite Rods Using
Ultrasonic Wave Propagation
by
Vanea R. Pharr
Submitted to the Department of Mechanical Engineering in Partial Fulfillment of the
Requirements for the Degrees of Naval Engineer and Masters of Science in Mechanical
Engineering
ABSTRACT
Nondestructive Evaluation (NDE) is a branch of applied science that is concerned with assessing
the properties and serviceability of materials and structures without causing collateral damage
or depreciation. This study presents a detailed analysis of advanced composite rods (comprised
of two or more distinct axial sections of different materials) using theoretical ultrasonic NDE. In
anticipation of the high elastic moduli of the rods (relative to many metals) along their
longitudinal axes, a one-dimensional wave propagation analysis will be conducted. By analyzing
the propagation of ultrasonic waves in nondispersive media and the corresponding reflections
and transmissions at structural interfaces, assessments of interfacial debonding will be explored
and the presence of anomalous materials can be demonstrated. The resulting graphical
presentations will be compiled and should provide the basis for material characterizations and
assessments of structural integrity throughout the rods.
Thesis Supervisor: James H. Williams
Title: Professor of Applied Mechanics
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Table of Contents
Biographical Note..........................................................................................................................................9
Acknow ledgem ents.....................................................................................................................................10
1. Introduction ............................................................................................................................................
11
1.1 w hat is NDE?. ....................................................................................................................................
11
1.2 Benefits of NDE .................................................................................................................................
11
1.3 Categorization of NDE Processes ...................................................................................................
12
2. Ultrasonic NDE ........................................................................................................................................
13
2.1 History of Ultrasonic NDE ..................................................................................................................
14
2.2 The Physics of Ultrasonics .................................................................................................................
15
2.2.1 The Classical Wave Equation and Its General Solution ..........................................................
16
2.2.2 Transverse and Longitudinal Waves .....................................................................................
16
2.2.3 Reflection ...................................................................................................................................
17
2.2.4 Transmission ..............................................................................................................................
21
2.2.5 Superposition .............................................................................................................................
23
2.2.6 Dispersion...................................................................................................................................
25
2.3 Ultrasonic NDE Setup and Procedure.............................................................................................
27
3. Linear Analysis of W ave Propagation.................................................................................................
30
3.1 W ave Propagation in a Long, Thin Rod........................................................................................
31
3.2 Reflection and Transm ission Coefficients......................................................................................
33
3.3 Behavior of the Reflection and Transm ission Coefficients ............................................................
35
3.3.1 Behavior as the Two M aterials Converge ..............................................................................
35
3.3.2 Behavior Due to Large Variations in Cross-Sectional Area....................................................
36
3.3.3 Behavior Due to Variations in M aterial Properties ...............................................................
43
3.3.4 Equivalent Behaviors..................................................................................................................
53
3.4 Characteristic Im pedance..................................................................................................................54
3.5 Using the Characteristic Impedance to Characterize an Intermediate Region .............................
58
3.6 Using the Characteristic Impedance to Identify an Unknown Material at an Interface ................ 66
4. Conclusion...............................................................................................................................................67
4.1 Derivation M ethodology ...................................................................................................................
67
4.2 M aterial Characterization .................................................................................................................
68
4.3 Areas for Further Study .....................................................................................................................
68
69
Appendix A : Table of M aterial Properties..............................................................................................
Appendix B: Material Characterization using Characteristic Impedance: Table and Plots.....................71
Material Characterization Table for Unknown Propagating Material of Incident Waveform ........... 72
Reflection Coefficients of Interface from Generic Aluminum Alloy to Various Common Materials........73
Transmission Coefficients of Interface from Generic Aluminum Alloy to Various Common Materials... 74
Reflection Coefficients of Interface from Beryllium Copper to Various Common Materials...............75
Transmission Coefficients of Interface from Beryllium Copper to Various Common Materials..........76
Reflection Coefficients of Interface from Boron Epoxy to Various Common Materials ......................
77
Transmission Coefficients of Interface from Boron Epoxy to Various Common Materials .................. 78
Reflection Coefficients of Interfacefrom Generic Cast Iron Alloy to Various Common Materials..........79
Transmission Coefficients of Interface from Generic Cast Iron Alloy to Various Common Materials.....80
Reflection Coefficients of Interface from Glass to Various Common Materials..................................81
Transmission Coefficients of Interface from Glass to Various Common Materials .............................
82
Reflection Coefficients of Interface from Graphite Epoxy to Various Common Materials..................83
Transmission Coefficients of Interface from Graphite Epoxy to Various Common Materials.............84
Reflection Coefficients of Interface from Nickel Steel to Various Common Materials.........................85
Transmission Coefficients of Interface from Nickel Steel to Various Common Materials...................86
Reflection Coefficients of Interface from Poly (methyl methacrylate) to Various Common Materials... 87
Transmission Coefficients of Interface from Poly (methyl methacrylate) to Various Common Materials
................................................................................................................................................................
88
Reflection Coefficients of Interface from Stainless Steel to Various Common Materials ................... 89
Transmission Coefficients of Interface from Stainless Steel to Various Common Materials .............. 90
Reflection Coefficients of Interface from Titanium to Various Common Materials ............................
91
Transmission Coefficients of Interface from Titanium to Various Common Materials .......................
92
Appendix C: Tabulated Reflection and Transmission Coefficients for Pairings of Selected Common
M a te rials .....................................................................................................................................................
93
Appendix D: Visual Representations of Reflections and Transmissions at Selected Material Interfaces 107
Appendix E: Expanding the W ave Function Argum ents ...........................................................................
113
E.1 Problem Definition ..........................................................................................................................
113
E.2 Deriving the Reflection and Transm ission Coefficients ...................................................................
114
E.3 Expanding the Wave Function Argum ents......................................................................................115
E.4 Exam p les .........................................................................................................................................
1 16
Appendix F: The Im portance of the Coupling ...........................................................................................
129
Spatial D ilation and Contraction ...........................................................................................................
129
M echanical Attenuation and Am plification ..........................................................................................
B ib lio g ra p h y ..............................................................................................................................................
133
13 5
Table of Figures
Figure 1: Longitudinal vs. transverse w aves in a rod. ............................................................................
17
Figure 2: W ave reflection in a rod at a fixed boundary. ........................................................................
18
Figure 3: W ave reflection at fixed and free boundaries. ........................................................................
19
Figure 4: W ave reflection m odeled with virtual waves. ........................................................................
19
Figure 5: Reflection in a fixed-free rod with an interior defect.............................................................. 20
Figure 6: Rod with defect that is aligned with wave pulse propagation direction.................................21
Figure 7: W ave pulse transm ission at a boundary................................................................................... 22
Figure 8: (a - c) Constructive and destructive interference (Principle of Superposition).......................24
Figure 9: Fixed boundary interaction producing same behavior as destructive interference example..... 25
Figure 10: Fourier decom position of a square wave. ............................................................................
26
(image obtained from http://mathworld.wolfram.com/FourierSeriesSquareWave.html).................... 26
Figure 11: Dispersion of a Gaussian w ave packet................................................................................... 27
27
(image obtained from http://www.jick.net/~jess/hr/skept/GWP/)......................................................
Figure 12: Basic ultrasonic N D E setup. ...................................................................................................
28
Figure 13: Alternative ultrasonic NDE test setups ..................................................................................
30
32
Figu re 14 : Ro d e le m e nt...............................................................................................................................
Figure 15: Com posite rod w ith junction at x = 0..................................................................................... 33
34
Figure 16: W ave transm ission at a boundary .........................................................................................
Figure 17: Rod composed of segments with identical chemical makeup and different cross-sectional
36
a re a s ............................................................................................................................................................
Figure 18: Reflection Coefficient Variation with Cross-Sectional Area (with area differentials across 6
39
o rd e rs o f m ag n itud e). .................................................................................................................................
Figure 19: Reflection Coefficient Variation with Cross-Sectional Area (with area differentials across 4
40
o rd e rs o f m ag n itud e). .................................................................................................................................
Cross-Sectional
Area
(with
small
area
differentials
Figure 20: Transmission Coefficient Variation with
41
across 6 o rders of m agnitude) ....................................................................................................................
Figure 21: Transmission Coefficient Variation with Cross-Sectional Area (with small area differentials
42
across 4 o rders of m agnitude) ....................................................................................................................
Figure 22: Reflection Coefficient Variation with Primary Wave Speed (with wave speed differentials
45
across 6 o rders of m agnitude) ....................................................................................................................
Figure 23: Reflection Coefficient Variation with Primary Wave Speed (with small wave speed differentials
46
across 4 o rders of m agnitude). ...................................................................................................................
Figure 24: Transmission Coefficient Variation with Primary Wave Speed (with wave speed differentials
47
across 6 o rders of m agnitude). ...................................................................................................................
Figure 25: Transmission Coefficient Variation with Primary Wave Speed (with small wave speed
48
differentials across 4 orders of m agnitude)............................................................................................
Figure 26: Reflection Coefficient Variation with Young's Modulus (with modulus differentials across 6
49
o rd e rs o f m ag n itud e). .................................................................................................................................
Figure 27: Reflection Coefficient Variation with Young's Modulus (with small modulus differentials across
50
4 o rd e rs o f m ag nitud e ). ..............................................................................................................................
Figure 29: Transmission Coefficient Variation with Young's Modulus (with small modulus differentials
52
across 4 o rders of m agnitude). ...................................................................................................................
Figure 30: Variation of cross-sectional area mimics the behavior of a wave pulse at an interface between
tw o d iffe re nt m ate ria ls...............................................................................................................................
54
Figure 31: Reflection and Transmission Coefficient Variation with Changes in Impedance Ratio......57
Figure 32: Composite rod with intermediate region (in pulse-echo mode setup).................................58
Figure 33: Boundary interactions w ith sam ple display..........................................................................
59
Figure 34: Sample calculation of first boundary in composite rod....................................................... 61
Figure 35: Sample display showing elapsed time intervals between receipt of consecutive signals......... 63
Figure 36: Wave pulse paths (through-transmission mode calculation).................................................65
Figure 37: Comparison of through-transmission mode path lengths......................................................66
Figure 38: Composite rod with single known material/ unknown material interface. .......................... 67
Figure B 1: Valid schematics for use of material characterization table/ plots......................................71
Figure D 1: Com posite rod w ith internal interface...................................................................................107
Figure D 2: Norm alized incident w ave pulse. ...........................................................................................
108
Figure E139: Com posite rod w ith junction. ..............................................................................................
113
Figure E 1: Com posite rod w ith junction. .................................................................................................
113
Figure E 2: D iscrete incident w aveform ....................................................................................................
117
Figure E 3: Projected future position of discrete incident pulse in absence of boundary at x = 0...........117
Figure E 4: Resultant w aveform s (discrete exam ple). ..............................................................................
118
Figure E 5: Discrete waveform reflection in individual time steps. ..........................................................
119
Figure E 6: Discrete waveform transmission in individual time steps. .....................................................
120
Figure E 7: Continuous incident step function w aveform . .......................................................................
121
Figure E 8: Projected future position of continuous incident step function waveform in absence of
bo u n d a ry at x = 0. .....................................................................................................................................
12 1
Figure E 9: Resultant waveforms (continuous step function example)....................................................122
Figure E 10: Continuous step function waveform reflection in individual time steps..............................123
Figure E 11: Continuous step function waveform transmission in individual time steps.........................124
Figure E 12: Continuous incident sinusoidal w aveform ............................................................................
125
Figure E 13: Projected future position of continuous incident sinusoidal waveform in the absence of a
b o u n d a ry at x = 0. .....................................................................................................................................
12 5
Figure E 14: Resultant waveforms (sinusoidal waveform example).........................................................126
Figure E 15: Continuous reflected sinusoidal waveform in individual time steps....................................127
Figure E 16: Continuous transmitted sinusoidal waveform in individual time steps. .............................. 128
Figure F 1: Wave pulse incident upon coupling/ test piece junction........................................................130
Figure F 2: Coupling-to-test piece transm ission with spatial dilation.......................................................131
Figure F 3: Coupling-to-test piece transmission with spatial contraction. ...............................................
132
Biographical Note
LT Vanea Pharr is originally from Raleigh, NC and joined the United States Navy as an
enlisted Sailor in July 2003. In 2007, she applied for and was accepted into the Seaman to Admiral
21 (STA-21) commissioning program, the Navy's premiere enlisted-to-officer commissioning
source. She earned her undergraduate degree in Applied Mathematics with a minor in Physics
from Old Dominion University, Norfolk, VA, and was commissioned as an Ensign in May 2009.
As an enlisted Sailor, LT Pharr served onboard the USS Enterprise (CVN 65), at Norfolk
Aircraft Intermediate Maintenance Department (AIMD), and at Naval Air Station (NAS) Oceana
Sea Operational Detachment (SEAOPDET). Upon acceptance to the STA-21 program, she
transferred to the Hampton Roads NROTC Unit to complete her undergraduate degree and
earned her commission as a Surface Warfare Officer - Engineering Duty Officer (SWO-ED)
Option. LT Pharr completed her SWO Division Officer tour onboard the USS Kearsarge (LHD 3),
where she held Division Officer positions in the Combat Electronics, Engineering Auxiliaries, and
Main Propulsion (FWD, AFT, and Oil Lab) divisions, prior to reporting to the Massachusetts
Institute of Technology to begin a three-year graduate course of study in naval architecture and
mechanical engineering. After graduation from MIT, LT Pharr hopes to apply her Naval Engineer's
degree to extensive work within the Navy's ship maintenance community, overseeing the
maintenance, construction, and refurbishment of the nation's naval fleet.
LT Pharr's military awards and decorations include the Surface Warfare Officer (SWO)
and Enlisted Aviation Warfare Specialist (EAWS) pins, the Navy and Marine Corps Achievement
Medal, the Navy Good Conduct Medal and other individual and unit awards. LT Pharr has one
daughter, Anaya, age 9.
9
Acknowledgements
This challenging academic program and culminating thesis work could not have been completed without
several individuals whose continuous support, encouragement, and guidance have made this work
possible. The list included here is not exhaustive by any means, but I would like to express earnest thanks
to the following:
- To Dr. James H. Williams, whose academic guidance and advisement have been invaluable. I am
extremely grateful for the faith placed in me to complete this work in a timely and first-rate manner.
- To all of my parents: Vanessa Garrett, Brian and Michelle Pharr, and Antoinette Bailey-Nottingham.
Your encouragement, support, guidance, and assistance made it possible for to complete this program,
particularly as a single parent. I am eternally grateful.
- To two of the best friends a girl could ask for, Courtney Jones Law and Kyra Lassiter, for late night
laughs and lamentations about the demands of academia! Thank you for helping me set and manage
expectations, remain focused, and maintain my sanity in the midst of the storm.
- Lastly, to my beautiful little girl, Anaya: Every plan I make and every achievement I strive for is done in
hopes of making your life that much better. You are my motivation and my inspiration, because I always
strive to be someone worthy of your admiration and emulation. Until the day that you become a
mother yourself, you will never comprehend just how much I love you.
10
1. Introduction
1.1 What is NDE?
Nondestructive evaluation (NDE) refers to the evaluation of materials, components, or structures
without causing collateral damage. The terms nondestructive testing (NDT), nondestructive inspection
(NDI), and nondestructive examination (NDE) are often used rather interchangeably in applicable
literature. However, in practice, there are generally slight differences in the scope and ultimate purpose
of the work referred to by each of these terms. For the purpose of this thesis, the term "nondestructive
evaluation" (or just "NDE") will be used to encompass all forms of nondestructive evaluation, testing,
inspection and examination. This is appropriate because, in general, nondestructive evaluation carries the
broadest definition with regards to the types of work it includes.
1.2 Benefits of NDE
NDE is invaluable to any process that requires the use of mechanical parts because, by nature,
mechanical parts begin to break down over time and with use. As such, verification of their ongoing ability
to properly complete process tasks is necessary to ensure continued production. Any application which
uses mechanical parts carries an inherent requirement for some form of evaluation or testing to ensure
the quality and accuracy of the finished product. Bray details three axioms that generally describe the
relationship between serviceability of mechanical parts and the need for some form of evaluation[1]:
1.
All materials have flaws.
2.
Flaws do not necessarily make a material unfit for service.
3.
Detectability of flaws increase with size.
As alluded to by axiom 2, since flaws in a part do not necessarily make that part unfit for service, having
to disassemble or decompose a part to verify its functionality, either partially or fully, is counterproductive
in the event that the part is either unflawed or flawed but still functional. Furthermore, experience tells
us that verifying the functionality of a part or piece through repeated disassembly and reassembly actually
increases the likelihood of creating a flaw because these disassembly/reassembly sequences produce
added and unnecessary wear and tear on the part. Every time the part is taken apart, there is a risk of
damage to components or improper reassembly after the assessment is complete. In addition, the time
required to fully decompose and reconstitute complicated mechanical parts or processes could make this
approach extremely inefficient and costly since, in industry, the down time of any machine or process due
to maintenance or quality assurance translates directly to dollars down the drain!
11
The specific benefits of nondestructive evaluation are many in the business world and in industry.
Quality assurance and accuracy verification methods that do not require the decomposition or
disassembly of machinery are less disruptive to production cycle and waste fewer resources. Parts
suppliers will routinely use NDE processes to certify the accuracy and quality of the parts they supply to
other businesses and verify that the parts meet customer specifications prior to shipping. Likewise,
customers of mechanical parts and machinery will use similar processes to verify specifications upon
receipt. These common practices both cut down on costs due to rework and improve the suppliers'
reputations when they routinely supply parts and materials that are devoid of defects: a sound business
practice. Similarly, implementing NDE processes at the manufacturing stage can yield lower production
costs and/or weights because safety factors can often be reduced when proper performance can be
verified nondestructively at an early stage. NDE processes at the manufacturing stage also help maintain
the steady flow of production. The periods between scheduled maintenance can be lengthened when
NDE processes provide greater insight to the actual periods of proper functionality of machinery, and
some scheduled maintenance meant to help maintain parts and systems that are, in fact, defect free can
be eliminated. Routine or continuous NDE can be implemented in certain processes to help detect
developing problems in machinery, allowing them to be corrected before they escalate. Both of these
help decrease disruption to the production cycle. Overall, NDE leads to more efficient processes and
designs, lower costs, higher quality assurance, and smoother production cycles.
1.3 Categorization of NDE Processes
Nondestructive testing and evaluation is an extremely broad and diverse field of applied science,
and its activities apply across a variety of industries and scientific fields. Similarly, NDE processes can be
categorized in multiple ways.
One way that NDE processes are often categorized is "active" or "passive", depending on how the
particular method actually interacts with a sample piece to detect faults. Active NDE involves the
introduction of energy in some form onto or into the specimen. This would include processes such as
those utilizing magnetics, eddy currents, and radiography, as well as ultrasonic NDE. Passive NDE includes
processes such as visual inspection or the use of visually enhancing liquids to coat a surface. Passive
processes also include observing a component or structure in the "as-is" state, or under the influence of
some typical/ characteristic load. Even something as simple as taking a measurement of a random part
after a batch comes off the line to ensure compliance with customer specifications qualifies as a form of
passive NDE. [1]
12
Another way to categorize NDE processes specifies the depth inside a sample or test piece at
which a particular process can still reliably identify a fault. For example, eddy currents can only identify
flaws at or near the surface of a material. Similarly, any type of visual inspection or visually enhancing
coating will only reveal a discrepancy at the surface of a structure. Thus, these processes would be
referred to as surface or near-surface techniques. Methods such as ultrasonics or radiography are known
as volumetric techniques because they can identify material flaws deep within the body of a structure.
Volumetric methods are typically more expensive and more involved than surface or near-surface
processes, requiring more training prior to proper administration and more elaborate testing
configurations.
A third way to categorize NDE techniques is by the particular scientific principles upon which they
are primarily based. For example, electromagnetic NDE would include such techniques as eddy currents,
magnetic particle inspection, magnetovision, remote field testing, and magnetic flux leakage testing,
among others. Sometimes this type of categorization helps with quick and easy identification of methods
that will be most appropriate for a particular application because, depending on the individual
circumstances, some forms of NDE will prove extremely helpful and informative for a particular
application while others will yield little to no useful information.
2. Ultrasonic NDE
Ultrasonic NDE is an active, volumetric type of nondestructive testing and evaluation that uses
the known behaviors of high frequency mechanical waves propagating through solids to identify material
properties and flaws within a test piece. The term ultrasonic refers to the use of mechanical waves above
the frequency range of human hearing, which is approximately 20 Hz to 20 kHz.' Typical frequencies for
ultrasonic NDE are in the range of 0.1 MHz - 10 MHz, although frequencies can range as high as 15 MHz
or more.[2] Benefits of this particular technique include its sensitivity to both surface and subsurface
characteristics, as well as the depth of flaw detection possible within the body of a subject. Also, ultrasonic
NDE is a highly accurate method of evaluation that can provide very detailed images of internal flaws
depending on the equipment used, and the tests require minimal preparation of the sample. However,
ultrasonic NDE processes require highly trained technicians to administer the tests, which generally add
to the expense. Ultrasonic NDE can also be somewhat sensitive to the type of materials present in a
I This is the definition of the term ultrasonic, as opposed to the term supersonic, which refers to a particle or body
which travels at a speed faster than the speed of sound in a particular medium, generally air.
13
sample. This particular type of NDE would not generally be utilized on test pieces that are extremely rough
or irregular in shape, grossly inhomogeneous in material composition, or comprised of overly coarse,
grained materials (such as cast iron) which would have low sound transmission.
2.1 History of Ultrasonic NDE
The earliest intentional use of sound waves to achieve imaging and/or ranging can arguably be
traced to the early sonar systems of World War
1.2
Following the war, extensive research and
developments in the area first led to the application of sonography to the field of medical diagnosis, where
ultrasounds became widely utilized to identify and image gallstones, breast masses, and tumors, a practice
which still carries on today. At the same time, the defense regime placed a major emphasis on NDE and
defect detection. This represented an overall shift in focus and led to significant developments in the
detection of material flaws in mechanical structures. At the time, the goal was that any structures found
to possess any material flaws at all would be retired from service. This quickly proved to be unnecessary
and impractical, but initially all available technologies, including ultrasonics, x-rays, eddy currents, and a
variety of other methods were explored to reveal potential applications to the field of flaw detection and
NDE.
As early as 1928, Soviet scientist Sergei Sokolov proposed the use of ultrasonic transmissions to
detect material flaws in metals. However, his grand ideas outpaced the technology of the time and, while
he was able to demonstrate a through-transmission technique for flaw detection shortly thereafter, his
developments were not quite mature enough for wide-range, practical use. In the early 1930s, several
other researchers arose with their own takes on the ultrasonic testing technique. In 1931, German
scientist 0. Muhlhauser obtained a patent for his machine which used two transducers placed at either
end of a test piece to transmit and detect the ultrasonic energy imposed. The first U.S. patent involving
ultrasonics expressly for use as a nondestructive testing technique in solids was granted to Sokolov in
1937 for a similar experimental setup, although his also incorporated the extensive work he had added to
his body of research over the preceding decade involving advanced coupling techniques and impedance
matching.[3] Just five years later another important patent, entitled "Flaw Detecting Device and
Measuring Instrument", was granted to Dr. Floyd Firestone, a researcher at the University of Michigan.[4]
Firestone's advances were monumental because he first introduced pulsed ultrasonic testing, using the
The first recorded use of sound propagation for detection by humans dates all the way back to 1490 and is credited
to Leonardo da Vinci, who was said to have used a tube with one end submerged in water and the other end to his
ear to detect the passing of ships. However, the first patents for underwater echo ranging devices were filed for and
obtained in Britain and Germany, respectively, shortly following the tragic sinking of the Titanic.
2
14
echo principle. The following passage, taken directly from the opening paragraphs of the claims and device
description provided in the patent application, show that the basic principles and intent of modern NDE
are evident.
My invention pertains to a device for detecting the presence of inhomogeneities of density or elasticity in
materials. For instance, if a costing has a hole or crack within it, my device enables the presence of the flaw to
be detected and its position located, even though the flaw lies entirely within the costing and no portion of it
extends out to the surface. My device may also be used for the measurement of the dimensions of objects, and
is particularly useful in those cases where one of the faces to which the measurement extends is inaccessible.
For instance, the thickness of the wall of a hollow ball or propeller can be measured with my device.
The general principle of my device consists in the sending of high frequency vibrations into the part to be
inspected, and the determination of the time intervals of arrival of the direct and reflected vibrations at one or
more stations on the surface of the part.- If metal parts a few inches long are to be inspected, these reflections
will arrive a few millionths of a second after the direct vibration is sent out, and the technique provided by my
invention enables these small time intervals to be measured by means suitablefor use in production inspection.
The purpose of my invention is to provide a means and method for indicating the presence of inhomogeneities
of density or elasticity in materials, especially in those cases where these inhomogeneities are entirely
surrounded by a mass of material so that they are not directly accessible; such a means enables parts which
are supposed to be homogeneous
andfree from cracks or holes to be inspected to see if this condition is met. [5]
With the successful implementation of the echo principle to ultrasonic testing, the technique became
relatively quick and simple, requiring access to only one end of a specimen rather than both. Most modern
ultrasonic NDE tests are fundamentally based on the Firestone apparatus, which he named the
"Reflectoscope" to highlight its use of reflected waves to detect discontinuities.
2.2 The Physics of Ultrasonics
Ultrasonic NDE utilizes the transmission of mechanical (sound) waves through a medium to
identify and characterize discontinuities. The discussion of the underlying physical principles provided
here is intended to be very basic and is only included to provide a context for the remainder of the
derivations presented throughout the body of this text. For a more in depth discussion of the physics of
sound waves and wave phenomena, several of the references cited provide excellent resources for
interested readers.
15
2.2.1 The Classical Wave Equation and Its General Solution
The classical wave equation is a second order partial differential equation and is given by:
a2((xt)
at
2
2 a
C
2
(Eq 1)
(Xt)
2
ax
where k(x,t) is a generalized coordinate which represents the displacement or motion, due to wave
propagation, of an infinitesimal element of mass in the medium of interest whose equilibrium position is
"x", and "c" is the speed of propagation of the wave and is wholly dependent on the properties of the
medium.
In the case of a one-dimensional wave, which will be the focus of this thesis, the solution to the
wave equation is well known, thanks to the work of d'Alembert and Euler3, and is given by:
(Eq2)
xt) = f(x -Ct) + g(x +ct)
Represented in this form, the wave can be thought of as being divided into two parts, propagating in
opposite directions, wheref and g are arbitrary functions of the argument x
ct, which is often called the
phase of the wave. Since the functionsf and g are indeed arbitrary, it is fully possible for one of them to
vanish completely, yielding a wave which fully propagates in one direction or the other entirely. However,
Eq 2 is the most general expression of the solution.
2.2.2 Transverse and Longitudinal Waves
When a waveform is incident upon a given medium, wave propagation occurs because of the
successive disruption of adjacent molecules as the wave's energy passes from one molecule to the next.
An exaggerated visual representation of the phenomenon is shown in Figure 1, where a hammer (or some
other instrument) is used to generate a wave pulse in a long, thin rod with one end fixed.
I French mathematician Jean-Baptiste le Rond d'Alembert is generally credited with having solved the onedimensional wave equation. His solution, which has since been coined "d'Alembert's solution" was first devised and
published in 1747 in the Memoirs of the Prussian Academy. However, his solution was limited by his
oversimplification of the boundary conditions of the problem, and so it did not properly describe observed
phenomena. This issue was later remedied by Swiss mathematician, Leonhard Euler, who expanded d'Alembert's
solution into a much more useful result.
16
(a)
(b)
Figure 1: Longitudinal vs. transverse waves in a rod.
In Figure 1 (a), the generated waveform is an example of a longitudinal wave because the displacements
resulting from the passing of the wave's energy are in the same direction as the wave propagation. More
specifically, a molecule that is disturbed by the passing of the wave pulse will oscillate back and forth
along the length of the rod, in the same direction that the wave is traveling. The opposite of this
description would be a transverse wave, shown in Figure 1 (b). In this case, the particle displacements
resulting from the disturbance are perpendicular to the direction of wave travel. A familiar example of a
transverse wave would be a plucked guitar string. If the vibrations of the string were significantly slowed
down in time, so as to be fully observable by the human eye, we would see that the individual molecules
oscillate up and down as the wave travels along the length of the string.
2.2.3 Reflection
The second important observable phenomena to discuss in wave propagation is reflection.
Consider the rod shown in Figure 2. Assuming that the composition of the rod is uniform throughout, the
wave pulse travels through the rod, uninterrupted, until reaching the far end, which is fixed. Upon
reaching the fixed end the wave pulse can no longer propagate in its original direction. At this point it will
reflect and travel back through the rod toward the end from which it originated. Neglecting the effects of
any damping or absorption as it travels (these effects will generally be ignored throughout this thesis), the
17
wave pulse will return to its point of origin where, with proper instrumentation, it could be measured and
observed.
t -to
-
1
-
-
-~
--
~
-
-
--
-
--
-
t =tj
--.
Figure 2: Wave reflection in a rod at a fixed boundary.
The overall effect of reflection on an incident wave pulse depends on the type of boundary the
pulse encounters. If the wave pulse encounters a fixed boundary, the reflected wave pulse is inverted
when compared to the incoming pulse. This can be thought of as a direct result of Newton's third law,
which tells us that if the pulse propagation causes the attached medium to exert an upward force on the
fixed boundary, the boundary must exert an equal but opposite downward force on the propagating
medium, as illustrated in Figure 3. As such, the reflected wave is inverted. However, in the case of a free
end, the reflected wave is not inverted and only the direction of travel is changed.[6]
18
\V/
Figure 3: Wave reflection at fixed and free boundaries.
A. P. French offers another interesting way to think of the reflected waves, modeling them instead as a
propagating virtual wave originating from the opposite side of the fixed or free end.[6] This idea is
illustrated in Figure 4. Whether the reflection phenomenon is thought of in this way or in terms of
Newton's third law, the result is equivalent.
Fixed Boundary
Free Boundary
Figure 4: Wave reflection modeled with virtual waves.
19
Now consider the rod of Figure 5, which contains a small defect in its interior. The defect will also
cause reflection of an incident wave pulse, similar to the boundaries of the previous example. However,
if the rod length and material properties (and, thus, the wave speed) are known, an observer taking
measurements and awaiting the pulse reflection would already know when to expect the reflected pulse
at the point of origin, assuming a homogeneous, defect-free sample interior. Hence, any reflections
measured after a shorter time interval has elapsed indicate the presence of a discontinuity or defect
within the rod which warrants further investigation. With a known wave speed, a simple time-speeddistance calculation will reveal the location of the defect within the interior of the rod. Additionally, with
certain types of instrumentation and appropriate imaging equipment, an image of the defect may be
produced. This idea represents the basic underlying principle of ultrasonic NDE.
t
to
t =t
Figure 5: Reflection in a fixed-free rod with an interior defect.
One potential shortfall of ultrasonic NDE can occur when the defect aligns with the wave pulse
such that the pulse is not reflected by the defect and continues to propagate, as illustrated in Figure 6. In
these instances, either another NDE method or an alternatively oriented wave pulse would be necessary
to reveal the presence of the defect. Luckily, there are many options for orienting the generated wave
pulse or for the general ultrasonic NDE setup. These will be discussed briefly in Section 2.3.
20
t =to
I
tt 2
Figure 6: Rod with defect that is aligned with wave pulse propagation direction.
2.2.4 Transmission
Now, suppose that the boundary encountered by the wave pulse is neither fixed nor free, but
instead it is an interface between two dissimilar materials. In this case the pulse is permitted to move past
4
the boundary and is transmitted into the new propagating medium.
Perhaps the first thing the propagating wave will "notice" about its new environment is the
change in material properties, which will result in a different wave speed. The mismatch in material wave
speeds leads to a change in the wavelength of the transmitted wave with respect to the incident wave.
To understand this, assume that the traveling wave pulse of Figure 7 takes a time period At to fully
transmit through the boundary at x = 0. In that time interval, the front end of the pulse will have travelled
a distance Axfront = c2 x At from the boundary and into the second medium. (Here, the subscript of c2
specifies the wave speed of the second medium.) However, the tail end of the pulse, still in the original
propagating medium, will have travelled a distance Axtaji = cl x At.
It should be apparent that, depending on the relationship between ci and c2, the front end of the
wave pulse will travel more
(c2 >
ci) or less
(c2
< c1) physical distance in the same time interval, At. Thus,
I When a wave pulse encounters a boundary between two dissimilar materials, part of the pulse is transmitted into
the new medium and the remainder is reflected. The proportions between the transmitted and reflected parts will
depend on the relative material characteristics between the two media, and the total energy of the incident wave
pulse is conserved.
21
the wave pulse is effectively stretched or compressed with the transmission. Figure 7 depicts wave
transmission into a medium with affaster wave speed (c 2
Medium 1:
E,, A 1, c,
>
ci), so the wave pulse is stretched in space.'
Medium 2:
I
E 2, A 2 -,C
2
t=t,
x
X=
0
t =t
X=
2
0
I
2
t=t,
x
x= 0
Figure 7: Wave pulse transmission at a boundary.
In addition to the wavelength of the pulse, transmission into a new medium also effects the
pulse's amplitude. To illustrate why this happens, imagine a transverse wave pulse traveling along a cord
of mass per unit length pi, which has been seamlessly spliced with another cord of mass per unit length
P2. From Newton's laws it is evident that when the wave encounters the interface, the first cord exerts an
upward force on the second. (Otherwise the two cords would not remain connected, which will be a
necessary boundary condition in analyzing problems of waves encountering a boundary.) However,
because of the mismatch in unit mass between the two cords, the same force from the first cord causes
different accelerations of the incremental cord lengths (i.e. masses) of the second cord. As such, the
resulting displacements in the second cord are greater (p2 < pi) or smaller
(P2 >
pi) and the amplitude of
s For clarity of the drawing, the reflected wave pulse is omitted in order to expressly highlight the effects of
transmission on the spatial characteristics of the wave pulse. However, all boundary interactions will necessarily
result in a reflected wave pulse, although it may be quite small in amplitude depending on the particular material
characteristics of the two propagating media.
22
the transmitted wave changes accordingly. The exact mathematical expression for the relative change in
the wave pulse amplitude will be derived later and referred to as the transmission coefficient.
2.2.5 Superposition
So far, this discussion of wave phenomena has focused on what happens when a wave pulse
comes in contact with some type of boundary: rigid (fixed), free, or an interface between two dissimilar
materials. But what happens when a wave pulse encounters another continuous wave or discrete wave
pulse? In this case, the two (or more) waves will interfere with one another and their combined energy is
summed together and acts upon the propagating medium via another phenomenon known as
superposition.
Consider the traveling waves shown in Figure 8 (a -c). In each case, the two waves come in contact
with one another and the resultant wave is simply the sum of the two individual pulses, as dictated by the
Principle of Superposition. In cases (a) and (b), constructive interference results in the aggregate wave
pulse having a greater amplitude than either of the original pulses. In case (c), destructive interference
results in the net cancellation of wave motion when the two pulses meet.
23
<C
a)
CN>A
t =tI
/'A N
x
t = t2
/772A
x
<C
/N74A
/'KA
Ix
b)
2c
t =t,
'C
>
/VA
x
t =t 2
/N--2A
x
2c
CN>A
/\N7-A
t =t3
x
t =t,
A>
A
x
t t2
C--
x
t = tx
F
C
Figure 8: (a - c) Constructiveand destructive interference (Principleof Superposition).
24
Incidentally, if the waves in Figure 8 (c) encounter reflecting boundaries at the two ends of the
propagating medium, they will each travel back towards the center and destructively interfere with one
another to cancel the wave motion at the same point. Indeed, if they continue to propagate, reflect, and
interfere, the point x = xO where they cancel one another will always remain stationary (neglecting any
kind of frictional effects). As such, this point could actually be replaced with a fixed node without changing
any of the actual wave dynamics, as shown in Figure 9.[6]
Cx
t =t,
A
AX
t =t
2
x
C~
t t
A
A
Figure 9: Fixed boundary interaction producing same behavior as destructive interference example.
2.2.6 Dispersion
As the abstract of this thesis indicates, the derivation presented in the following sections is
concerned with nondispersive media, but in order to make this specification it is important to at least
define the phenomenon of wave dispersion, specify what it is, and briefly address why it occurs.
Until now it has been assumed that all mechanical waves traveling in a given medium will travel
with the same wave speed, defined solely by the medium's material properties. In practice, however, this
is not the case. In general, the wave speed in a given medium is a function of wavelength:
CP = cP(A)
(Eq 3)
More specifically, waves with shorter wavelengths tend to travel slower than waves with longer
wavelengths. Wavelength and wave speed are related by a dispersion relation, which can be defined
and/or stated in multiple ways depending on the type of wave discussed, the propagating medium, and
25
the preferred wave properties of choice.6 The relation can be linear, as in the case of electromagnetic
waves propagating in a vacuum, or nonlinear.
The name can be better understood through consideration of the mathematical representation
of a waveform. Utilizing the famous Fourier transform, any complex waveform can be decomposed into
simpler, trigonometric functions and expressed as an infinite sum.' An example of the Fourier
decomposition of a square wave is shown in Figure 10 (below). In this example, the colored trigonometric
functions each represent one function in the corresponding Fourier series, obtained from applying the
Fourier transform to the Heaviside step function. The series is, by definition, an infinite series and the sum
becomes an increasingly better approximation of the actual step function as more and more of the simpler
functions are added together.
f(x)
Figure 10: Fourier decomposition of a square wave.
(image obtainedfrom http://mathworld.wolfram.com/FourierSeriesSquare Wave. html)
From this example, it is apparent that a complicated waveform can be equivalently expressed
(mathematically) as a sum of simpler, trigonometric functions, or thought of (physically) as a wave packet
that contains lots of little waves, each of which may have a different wavelength. In practice, this means
that when a mechanical wave is incident upon a dispersive medium, each of the constituent wave
functions will be traveling at their own unique speeds based upon their individual wavelengths. As such,
some of the constituent waves will travel slower than others (those with shorter wavelengths) and others
k = 2r
6 Dispersion relations are commonly stated in terms of angular frequency, a> = 2wf, and wavenumber,
because both the group velocity and the phase velocity have convenient expressions in these variables.
7 The process of decomposing a function into its corresponding sum of simpler functions is known as Fourier
analysis. The opposite practice of recovering the more complex waveform from its constituent simple functions is
known as Fourier synthesis. Both processes, as well as the aforementioned transform, and the resulting Fourier
series, are named for the French mathematician, Jean-Baptiste Joseph Fourier.
26
will travel faster (those with longer wavelengths). This causes the original, incident wave to disperse, or
spread out in space, as the individual components separate from one another in the medium due to their
different wave speeds.
E
C
Figure 11: Dispersion of a Gaussian wave packet.
(image obtained from http://www.jick.net/~jess/hr/skept/GWP/)
The dispersion relation (and likely its derivation and a detailed explanation of its meaning) can be
found in virtually any reference that covers mechanical wave behavior. In practice, dispersion becomes
an increasingly important phenomenon in applications or observations that cover large distances and time
periods, when waves have the opportunity for large dispersion effects, such as ocean waves in
hydrodynamics. For the waves of practical ultrasonic NDE applications, the distances covered and
associated time periods are generally small enough that dispersion can be ignored without the
introduction of significant errors. As such, this thesis deals expressly with nondispersive materials and the
dispersion of waves is ignored.
2.3 Ultrasonic NDE Setup and Procedure
At its most basic, there are five primary parts to the ultrasonic NDE setup:
1.
Pulse generator
2.
Coupling
3.
Specimen (test piece)
27
4.
Receiver
5.
Display Unit (for appropriate data collection)
DISPLAY UNIT
PULSE GENERATOR
TEST PIECE
G
COUPLIN
G
RE4
RECEIVER
Figure 12: Basic ultrasonic NDE setup.
The pulse generator does exactly what its name suggests: generate the ultrasonic wave pulse that will
be used to take measurements and detect anomalies or defects. The typical method of pulse generation
uses a piezoelectric transducer, which is a device that converts electrical pulses into mechanical
vibrations. The generated disturbance can be comprised of individual pulses or it can be a continuous
wave, but generally the modern practice is to use individual pulses to avoid the development of standing
waves8 within the specimen, which would make useful measurements impossible.
In order to properly transfer the generated pulse into the specimen to be tested, some type of
coupling is necessary. This is because the acoustic impedance of the specimen (i.e. its resistance to
transmitting the disturbance) is much, much higher than that of air. Thus, without some type of coupling
to displace the air between the transducer and the specimen, nearly all of the energy in the generated
8 A standing wave, also known as a stationary wave, is a wave that remains in a constant position. This phenomenon
can occur
generated
relative to
the waves
when the medium of propagation moves relative to a propagating wave, or standing waves can be
as the result of interference between two waves of equal amplitude traveling in opposite directions
one another. In the latter case, the two interfering waves could be the incident wave and its reflection, if
are long enough in space (or continuous) to allow for ongoing interference.
28
pulse would be reflected off the specimen with very little energy actually transmitted into the specimen
body. As previously mentioned, a great deal of research has gone into optimizing the coupling between
the pulse generator and the specimen. The coupling chosen can have important effects on the entire
process (see Appendix F) and so the decision must be carefully weighed. Generally the coupling will be
some type of liquid. A thin film of oil, glycerin, or water is often used, or the NDE can be conducted in full
immersion, typically in water.[7]
Also coupled to the test piece is a receiver which will sense the wave pulse as it exits the specimen.
The receiver can be a separate piece of equipment from the transducer, as in through-transmission mode,
or a transceiver can perform the functions of both the transducer and the receiver, as in pulse-echo mode.
For proper measurement, the receiver will also be connected to an oscilloscope or some other means of
providing appropriate output display for the human interface.
A few examples of possible test configurations are sketched below, but any number of configurations
are possible. Often, multiple configurations or orientations will be used on the same test piece to ensure
circumstance such as those shown in Figure 6 do not occur. Also, the specific circumstances of the
specimen may dictate certain aspects of the configuration, such as instances where only one end of the
test piece is accessible, necessitating pulse-echo mode NDE.
29
--
I
:
j
S
\1/
&OP
q
I
Figure 13: Alternative ultrasonic NDE test setups.
3. Linear Analysis of Wave Propagation
The measurable quantities that contain the useful information garnered from ultrasonic NDE
processes, known as the reflection coefficient and transmission coefficient, can now be introduced. An
30
abbreviated derivation of these quantities, based in large part on the derivation presented in the
unpublished notes by Dr. James H. Williams of the Massachusetts Institute of Technology entitled "Wave
Propagation: An Introduction to Linear Analyses"[8], is presented in Appendix E. The final results of this
derivation will be utilized in later sections and in the appendices to motivate assessments of wave pulse
behavior and to develop graphical presentations which have been compiled to provide the basis for
material characterizations and assessments of the structural integrity of a test specimen. These graphical
presentations are, to the best of my knowledge, original work that is not currently presented, in whole or
in part, in any other source.
3.1 Wave Propagation in a Long, Thin Rod
For the nondispersive rod, the wave speed is known from the material properties and is defined
as c =
-,
p
where E is the material's modulus of elasticity and p is its density. Thus, in a rod, Eq 1 takes the
(Eq 4)
_ E a2((x,t)
az (x,t)
2
p ax 2
form:
at
This equation takes different forms in other continua because the wave speed is specified by the medium
in question and its particular properties.[8]
To understand the physics happening inside the rod due to wave propagation, consider an
-
dx
-, respectively.
31
+
infinitesimal element of mass within the rod whose width is dx and whose equilibrium position is centered
dx
at x. As such, the left and right limits of this elemental mass, at equilibrium, are located at x - 2 and x
&r
Vi
x-dx x xidx
2
2
Figure 14: Rod element.
When a wave pulse perturbs this elemental mass, the displacements of either side of the element
are given by 4(x -
, t) and
4(x
+ dx, t). Thus, the total change in length of the element is:
AL =
dx, t)
(X
-
(Eq 5)
, t)
The strain (e) in an element is defined as the change in length (in this case due to the perturbation caused
by wave propagation) divided by the unperturbed length, or:
f(Xx
E
,t)dx
x+2
(Xx
f(Xx
,t)
dx'
-
,t)-f
(x-
, t)
d
(Eq 6)
2
UM E =
dx->O
af (x,t)
(Eq 7)
Ox
The stress (a) in an element is defined as the force per unit area, and the relationship between stress and
strain is given by:
(Eq 8)
a= EE
Therefore, the elastic force in the elemental mass due to the perturbation caused by the wave pulse can
be found directly.
o =
A
= Ee-
F
= EAe = EA
OX
'O(xt)
This definition of the elastic force is utilized in the derivation presented in Appendix E.
32
(Eq 9)
3.2 Reflection and Transmission Coefficients
Consider the wave pulse of Figure 15, propagating in a semi-infinite rod which is bounded at
x = 0 by another semi-infinite rod.9 The two are joined by a seamless junction such that they always remain
in constant contact. The density, modulus of elasticity, cross-sectional area, and wave speed in the first
rod are given by pi, E 1, A1, and ci, respectively. Similarly, the material properties of the second rod are P2,
E2, A 2, and c2.
E1, A1 , c,
E 2 , A 2 , C2
X
x=O
Figure 15: Composite rod with junction at x = 0.
Upon encountering the interface between the two rod segments, the incident wave pulse's
energy will be divided between two resultant pulses: a reflected wave pulse and a transmitted wave pulse,
as previously discussed. This process is illustrated in Figure 16 below. Notice that the incident and
reflected wave pulses coexist in the original rod, at least momentarily and in the vicinity of the boundary,
while the transmitted wave pulse alone propagates in the second rod.
9
The placement of this boundary at x =
0 is arbitrary, so chosen for convenience.
33
Ell A, c,
E 2 , A 2, C 2
t=to
x
x
=0
El, A1 , c,
E21 A 2 , C 2
t=tl
x
x=O
Ell All c,
E 2, A 2, c 2
t =t2
X
k I
x =0
E2, A 2 ,
El lXA, cl
C2
t = t3
x =0
Figure 16: Wave transmission at a boundary.
34
The reflected and transmitted waves can be defined in terms of the incident wave.
fi(-x, t)
g,(xi, t) =
2E AIc
fE1 A 1 c 21 +E 2 2A 2 c 1
fi (
C1
2x ,
t)
(Eq 10)
(Eq 11)
Full derivations of Eqs 10 and 11 are provided in Appendix E. The amplification factors multiplied by the
incident wave function in the equations for the reflected and transmitted waveforms are given the fairly
self-explanatory names reflection coefficient and transmission coefficient, respectively. Appendix C
provides the tabulated values of the reflection and transmission coefficients of various material pairs,
consisting of the common materials whose properties are found in Appendix A.
In terms of ultrasonic NDE, these are the measurable quantities that provide information about
the interior body of a test specimen. During an actual ultrasonic NDE process, the amplitudes of the
reflected and/or transmitted waveforms can be physically measured and then compared with the known
amplitude of the incident wave. In this way, the reflection and transmission coefficients can be
determined experimentally and the propagating material(s) identified.
3.3 Behavior of the Reflection and Transmission Coefficients
The previous section provides the definition of the reflection and transmission coefficients:
=
E1 Ajc 2 -E 2 A 2 c1
(Eq 12)
T =
2E1 A1c2
E 1Aic 2 +E 2 A 2c 1
(Eq 13)
R
-
E1 Ajc 2 +E 2A 2C 1
In an effort to provide greater insight into their physical significance, let us briefly look at the behavior of
R and T in a few limiting cases.
3.3.1 Behavior as the Two Materials Converge
In the limit as E 1 -E
2,
A 1-A
2,
and
c14c2,
the reflection coefficient goes to zero while the
transmission coefficient goes to one. Physically, this limit is equivalent to a constant, homogenous rod
with the same material of propagation throughout. As such, there is no internal boundary to cause
reflection and no internal reflections occur. Similarly, if the rod is made of only one material, no changes
in the wave pulse properties occur due to transmission because there is no internal boundary for the wave
to traverse or a second material of propagation for the wave to transmit into. The wave observed at the
far end of the rod will remain unchanged from the initial, incident wave, with the exception of frictional
and absorption effects.
35
3.3.2 Behavior Due to Large Variations in Cross-Sectional Area
Consider the rod of Figure 17, where both segments consist of identical material but the crosssectional areas vary greatly.
EA 1 ,c
E, A 2 c
x
0x
Figure 17: Rod composed of segments with identical chemical makeup and different cross-sectional areas.
Let A 2 = nA, where A is the cross-sectional area of the first segment and n is allowed to take the value of
any real number over a broad range of values. From Eqs 12 and 13, the reflection and transmission
coefficients now reduce
to
36
R=
EAc-E(nA)c
EAc+E(nA)c
=
T=
T
2EAc
EAc+E(nA)c
_
1-n
(Eq 14)
1+n
(Eq 15)
1+n
_2
From Eq 14, it is apparent that the reflection coefficient converges to 1 when the segment beyond
the boundary has a very small cross-sectional area compared to that of the originating piece (n <<1) and
converges to -1 as n increases and becomes much larger than one, or as the cross-sectional area of the
segment beyond the boundary becomes large compared to that of the originating piece. This is shown
over a large range of cross-sectional area differentials in Figure 18. The reflection coefficient Rf
approaching a limiting value of one for n << 1 is indicative of the behavior of a propagating wave upon
reaching a free end boundary. In this instance, the cross-sectional area of the second segment becomes
so small as to make the second segment virtually invisible to the propagating wave, and the wave reflects
almost as though it had encountered a free end. Conversely, the reflection coefficient converges to -1 as
n increases, which indicates the behavior of a propagating wave as it encounters a rigid boundary. Indeed,
if the cross-sectional area of the segment following the boundary is 100 or 1000 times greater than that
of the originating segment, the second segment effectively acts as a wall, or a rigid boundary. From Figure
19, which shows the lower ranges of n more clearly, the graph is seen to cross the x-axis (R = 0) at n = 1.
This is equivalent to one continuous rod of constant cross-sectional area, as previously discussed.
As n
-
0, the limit of the transmission coefficient is large (T
-+
2), and the values converge to zero
as n becomes much larger than one, as shown in Figure 20. At the lower limit, this behavior indicates that
the end of the first medium of propagation closely approximates a free end boundary, as previously
discussed and seen in the behavior of the reflection coefficient. Because the second segment has a very
small unit mass due to its small cross-sectional area, any transmitted energy will exert relatively large
forces on the elemental masses of the second rod segment, resulting in large accelerations of the
elemental masses of the second segment and equally large displacements. As n becomes large, the
transmission coefficient converges to zero, which is consistent with the earlier analysis that, in this limit,
the second segment effectively acts as a rigid boundary and transmission does not occur.
The more detailed plot of the lower ranges of n values presented in Figure 21 shows that the
transmission coefficient equals one for n = 1, as expected. Since there is effectively "no boundary" when
both segments have approximately the same cross-sectional area, we would expect the transmitted wave
37
to be largely unchanged from the incident wave in the vicinity of n = 1 (such as n = 0.99 or n = 1.01), and
very, very little of the wave's energy to be reflected (R =-: 0 in the vicinity of n = 1).
38
Reflection Coefficient Variation with Cross-Sectional Area Differential
(6 Orders of Magnitude: 10-3 to 103)
1.5
1
0.5
U
Cross-Sectional Area Multiplicative Factor, n = A2/Aj
0
U
C
0
0
200
-200
600
400
-0.5
-1
-1.5
Figure 18: Reflection Coefficient Variation with Cross-Sectional Area (with area differentials across 6 orders of magnitude).
39
800
1000
1200
Reflection Coefficient Variation with Cross-Sectional Area
(4 Orders of Magnitude: 10- to 10)
1.2
1
0.8
0.6
0.4
0.2
Cross-Sectional Area Multiplicative Factor, n =A2/A
0
C
S
-.
0
0
-
02
T
--------
--
-
--
6
4
8
-0.2
-0.4
-0.6
-0.8
-1
Figure 19: Reflection Coefficient Variation with Cross-Sectional Area (with area differentials across 4 orders of magnitude).
40
10
12
Transmission Coefficient Variation with Cross-Sectional Area
Differential
(6 Orders of Magnitude: 10-3 to 103)
2.5
2
c)
1.5
0
U
C
0
E
Lin
C
1
I-
0.5
0-200
0
200
600
400
800
Cross-Sectional Area Multiplicative Factor, n = A 2/A 1
Figure 20: Transmission Coefficient Variation with Cross-Sectional Area (with small area differentials across 6 orders of magnitude).
41
1000
1200
Transmission Coefficient Variation with Cross-Sectional Area
Differential
(4 Orders of Magnitude: 10-3 to 10)
2.5
2
1.5
a)
0
.0
0.5
-
-0
2
-
-
0
6
4
8
Cross-Sectional Area Multiplicative Factor, n = A 2/A 1
Figure 21: Transmission Coefficient Variation with Cross-Sectional Area (with small area differentials across 4 orders of magnitude).
42
10
12
3.3.3 Behavior Due to Variations in Material Properties
Also present in the definitions of the reflection and transmission coefficients are the elastic moduli
and characteristic wave speeds of the two propagating media. Making similar representations of the
modulus of elasticity and wave speed of the second medium, namely E 2 = mE and C2 = nc, and specifying
that the two media share the same cross-sectional area A, expressions similar to Eqs 14 and 15 can be
developed, analyzed, and plotted to observe the behavior of the reflection and transmission coefficients
with large variations in these material properties. Here, the properties of the first medium are simply
denoted E and c for simplicity.
R
T=
=
EA(nc)-(mE)Ac _ n-m
EA(nc)+(mE)Ac
n+m
T
2EA(nc)
EA(nc)+(rE)Ac
(Eq 16)
-2n
n+m(Eq
17)
Allowing both values to vary simultaneously over any significant range yields a three-dimensional
plot with far too much rapid variation and few (if any) discernable points or features of interest. However,
holding each variable, in turn, to a constant value (such as n,m = 1) while the other is allowed to vary over
a large range produces two-dimensional plots similar to those in the previous section and provides some
insight as to the behavior of the reflection and transmission coefficients as these material properties vary.
The plots are slightly deceptive because these two material properties are related rather than
being fully independent of one another. After all, wave speed is defined in terms of the elastic modulus
and the material density. Also, there are general trends with regard to material density and elasticity.
Materials with higher densities tend to be stiffer, and thus have higher elastic moduli, since the elastic
modulus is a measure of a material's resistance to deformation in the elastic region. However, there are
clear exceptions to this trend (see the Material Properties data in Appendix A) and it is far from a hard
and fast rule. Similarly, with all else held equal, wave speed and density have an inverse relationship (recall
the definition of material wave speed from Section 3.1 and Eq 4), so higher primary wave speeds indicate
lower material densities when the elastic modulus is held constant. Nonetheless, the behaviors exhibited
in the plots give some intuition as to the physical mechanisms governing interactions at boundaries
between two materials with very similar elastic moduli but noticeably different wave speeds (such as
nickel steel and boron epoxy), or vice versa.
As shown in Figure 22, materials with similar elastic moduli but different wave speeds produce a
reflection coefficient that trends rapidly to 1 as the wave speed of the second material increases relative
43
to that of the first. This result is consistent with the behavior of a wave pulse reflecting at a free end. Since
denser materials will have a lower principal wave speed (for the same modulus of elasticity) and vice
versa, the high wave speed of the second material indicates a less dense material which closely
approximates a free end in wave reflection. Similarly, at the opposite limit, the extremely low relative
wave speed of the second material indicates a highly dense material which behaves like a fixed boundary
and the reflection coefficient tends to -1. From the complementary plots of transmission coefficient
variation with changes in the primary wave speed (Figures 24 and 25), we see that the coefficient of a
wave transmission from a material with a relatively high primary wave speed to a material with a much
lower primary wave speed (n << 1) looks like that of a clamped boundary transmission and the
transmission coefficient goes to zero. The opposite limit closely approximates a free end boundary and
the transmission coefficient is high (T
-
2), showing large displacements in the second, less dense
medium, whose low elemental mass allows for its molecules to be greatly affected by the energy of the
wave pulse.
From Figure 26, we see that as the elastic modulus of the second material increases relative to
the first, which equates to an extremely inelastic second material relative to the first, the reflection
coefficient converges to -1. This behavior is consistent with a wave pulse reflecting at a rigid boundary.
This occurs because, in the limit of an extremely large difference between the two materials' elasticity, an
extremely inelastic material would function like a rigid boundary when compared to a more elastic
material of incidence, reflecting almost all of the pulse's energy ( I RJ - 1) and inverting the pulse like a
clamped end would (Figure 3). At the opposite extreme (an extremely inelastic material of incidence
interfaced with a highly elastic second material), the reflection coefficient converges to 1, because the
more elastic material is free to move compared to the inelastic material of incidence, and the interface
acts effectively as a free end. The behavior illustrated in the plots of the transmission coefficient with
variation of the elastic modulus (Figures 28 and 29) are consistent with these results as well. When the
elasticity of the second material is much greater than that of the first (m << 1), the transmission behavior
mimics a free end and large displacements are observed in the second material. Similarly, when the
second material is extremely inelastic compared to the first, the second material behaves as a fixed
boundary. Thus, very little wave energy is transmitted and the transmission coefficient goes to 0.
44
Reflection Coefficient Variation with Primary Wave Speed Differential
(6 Orders of Magnitude: 10-3 to 103)
1.5
1
0.5
U
Primary Wave Speed Multiplicative Factor, n
0
U
=
C2/C1
-G
200
c0-200
600
400
800
a)
a),
-0.5
-1
-1.5
Figure 22: Reflection Coefficient Variation with Primary Wave Speed (with wave speed differentials across 6 orders of magnitude).
45
1000
1200
Reflection Coefficient Variation with Primary Wave Speed Differential
(4 Orders of Magnitude: 10-3 to 10)
1
0.8
0.6
0.4
0.2
Primary Wave Speed Multiplicative Factor, n = c2 /ci
0
0
U
2
6
4
8
c
-.0 -0.2
-0.4
-0.6
-0.8
-1
-1.2
Figure 23: Reflection Coefficient Variation with Primary Wave Speed (with small wave speed differentials across 4 orders of magnitude).
46
10
12
Transmission Coefficient Variation with Primary Wave Speed
Differential
(6 Orders of Magnitude: 10-3 to 103)
2.5
2
1.5
(U
0
U
C
0
1
E
Li,
C-
0.5
0
0
-200
200
600
400
800
Primary Wave Speed Multiplicative Factor, n = c2/C1
-0.5
Figure 24: Transmission Coefficient Variation with Primary Wave Speed (with wave speed differentials across 6 orders of magnitude).
47
1000
1200
Transmission Coefficient Variation with Primary Wave Speed
Differential
(4 Orders of Magnitude: 10-3 to 10)
2
1.8
1.6
1.4
.
1.2
0
U
0
E
Ln
0.8
0.6
0.4
0
-
-
0.2
2
0
6
4
8
Primary Wave Speed Multiplicative Factor, n = C2/C
Figure 25: Transmission Coefficient Variation with Primary Wave Speed (with small wave speed differentials across 4 orders of magnitude).
-u
-
-
48
10
12
Reflection Coefficient Variation with Young's Modulus Differential
(6 Orders of Magnitude: 10-3 to 103)
1.5
1
0.5
0
U-
0
C
200
5o- 20 0
600
400
4)
-0.5
-1
-1.5
Young's Modulus Multiplicative Factor, m = E 2/Ej
Figure 26: Reflection Coefficient Variation with Young's Modulus (with modulus differentials across 6 orders of magnitude).
49
800
1000
1200
Reflection Coefficient Variation with Young's Modulus Differential
(4 Orders of Magnitude: 10-3 to 10)
1.2
1
0.8
0.6
0.4
U
t0.2
0
U
00
U
2
12
10
8
6
0
~ 0.2
-0.4
-0.6
-0.8
-1
Young's Modulus Multiplicative Factor, m =E2/Ej
Figure 27: Reflection Coefficient Variation with Young's Modulus (with small modulus differentials across 4 orders of magnitude).
50
Transmission Coefficient Variation with Young's ModulusDifferential
(6 Orders of Magnitude: 10-3 to 103)
2.5
2
Q)
1.5
0
U
C
0
E
Ln
1
0.5
0
-200
0
200
600
400
800
Young's Modulus Multiplicative Factor, m = E2/Ej
Figure 28: Transmission Coefficient Variation with Young's Modulus (with modulus differentials across 6 orders of magnitude).
51
1000
1200
Transmission Coefficient Variation with Young's Modulus Differential
(4 Orders of Magnitude: 10-3 to 10)
2.5
2
1.5
0
U
0
'4-
0.5
0
0
--2
6
4
8
Young's Modulus Multiplicative Factor, m =E2/Ej
Figure 29: Transmission Coefficient Variation with Young's Modulus (with small modulus differentials across 4 orders of magnitude).
52
10
12
3.3.4 Equivalent Behaviors
From the plots presented in the previous sections, it is interesting to note different material
variations that produce equivalent behavior in the reflection and transmission coefficients with equivalent
differentials between the two rod segments. For example, varying the modulus of elasticity of the second
segment with respect to the first while holding the wave speed and cross-sectional area constant
throughout the composite rod produces the same behavior in the reflection and transmission coefficients
as varying the cross-sectional area of the second segment with respect to the first while holding the wave
speed and modulus of elasticity of the rod constant throughout (compare Figures 18 and 20 with Figures
26 and 28). The table below summarizes all of the equivalent behaviors that can be observed.
Cross-SectionalArea
Equivalent Variations of
Reflection Coefficient
Transmission Coefficient
Variation Expression
Material Properties
Expression
Expression
2
1-n
A 2 =nA1
E2 =nE1 or c1 =nc2
A 1 =nA 2
c2 =nc1 or E1 =nE 2
R_= T
-
R =
n-i
T =
2n
Note that, while the mathematical expressions depend upon whether the characteristics of the
first rod segment are varied with respect to the second or vice versa, the physical results of either scenario
are captured within any one set of plots of the relevant variation presented in the previous sections
because of the wide range of values applied to the multiplicative factors in each expression. For example,
the substitution A 2 = nA, yields the expressions for the reflection and transmission coefficients stated in
Eqs 14 and 15. However, the opposite substitution, namely A, = mA 2, represents behavior that is still
captured in the plots of Figures 18 - 21 because the multiplicative factor n is allowed to vary over a large
range. Mathematically, m = 0.1 is equivalent to n = 10, both of which are shown on the aforementioned
plots. As such, the important observation is that equivalent behaviors can be obtained with appropriate
variations of any one of these three characteristics when they are varied independently of the other two.
More specifically, the behaviors of the reflection/ transmission coefficients as a result of variations in
material properties can be mimicked with variations in cross-sectional area.
These equivalent behaviors might be applied to observe the behavior of ultrasonic waves at an
interface between two particular materials when one of the materials of interest is not readily available.
For example, from the material properties presented in Appendix A, the primary wave speed of boron/
epoxy is approximately twice that of nickel steel, and their moduli of elasticity are very close. If nickel steel
was readily available but boron/epoxy was not (or vice versa), an experimenter could observe
53
approximately the same response of an ultrasonic wave pulse at an interface between these two materials
by creating a composite rod composed only of nickel steel and making appropriate variations of the crosssectional areas of the two rod segments. Similarly, the response of a wave pulse at an interface between
glass and cast iron can be observed with only one of the two materials and an appropriate variation of
cross-sectional area because of their nearly equivalent primary wave speeds.
Boron/ Epoxy
Nickel Steel
0
MEMOMM"Efts1000ft1w.
Nickel Steel
Nickel Steel
FW-
A,= 2A 2
Figure 30: Variation of cross-sectional area mimics the behavior of a wave pulse at an interface between two different materials.
3.4 Characteristic Impedance
Recalling the definition of the wave speed (Section 3.1), we can introduce a quantity known as
the characteristic impedance which allows for additional comparisons between the material properties of
the two propagating materials and some qualitative analysis of the effects of any disparities.
Traditionally, the term "impedance" is associated with electrical systems. It is defined as the ratio
of the driving voltage to the current. Translating electrical quantities into mechanical ones, a similar ratio
can be applied to mechanical systems. For a mechanical system, specifically a rod acting as the
54
propagating medium for a mechanical wave, the voltage (which, in the electrical system, is the driving
force of the motion) equates to the force of the propagating wave, which drives the displacements of
elemental masses in the medium. Similarly, the current, which is the velocity of individual charges in an
electrical system, equates to the velocity of the oscillations of particles in the rod due to wave
propagation.[8] With this analogy, the characteristic impedance, Z, can be defined:
Z=
Driving Force
ParticleVelocity
(Eq 18)
EA
ia
at
The spatial partial derivatives of the wave function are derived in Appendix E, and the temporal partial
derivatives can be similarly derived using the chain rule. Making the appropriate substitutions into Eq 18
yields the following expression for characteristic impedance:
|EA
EAff(x-cpt)
EA
-cof(x-cyt)
cp
(Eq 19)10
This quantity is given the name characteristic impedance because it is characteristic of a particular
material, since it is based on material properties. Using this definition of characteristic impedance, the
reflection and transmission coefficients can be rewritten.
1
R =
E 1A1 c 2 -E
c
A
_ E1 A 1 c 2 -E 2 A 2c 1 c
E1 A 1 c 2 +E 2 A 2c 1 I
E1 Aic 2 +E22 A 22c 11
2
ClC 2
_CZI-Z 2
Z1+Z2
(Eq 20)
_c 2Z1
(Eq 21)
1
2E 1 A 1 c 2
= E1 Alc 2 +E 2 A 2 cl
_
2EAlc
c 2
E1 AIc 2 +E 2 A 2 cI 1
(Eq+Z2
Now, a comparison between the two materials can be made by considering a ratio of their characteristic
impedances. Since this quantity is a derived quantity, comprised of the material's density, modulus of
elasticity, and cross-sectional area, a direct comparison of just one of these properties is not readily
possible. However, comparisons between characteristic impedances imply certain possible comparisons
between the constituent properties. For example:
Z1 >> Z2
E 1AI >
C1
E 2A 2
C2
Recall that, by definition, cp is a positive value because it is defined as a square root. Similarly, modulus of
elasticity and cross-sectional area are positive quantities as well, so Z> 0 and the absolute value can be dropped.
10
55
.- >> 1 Z2
Thus, the comparison
Z2
1A
2
E2 A2 C 1
>>
LLL
(Eq
22)
(E2)
>> 1 implies either El >> E 2, A >> A 2, C2>> c1, or a combination of the three. Plots
showing the behaviors of the reflection and transmission coefficients over a range of the ratio between
characteristic impedances are shown below. Notice that in the limit
KK 1, the behavior of the reflection
Z2
and transmission coefficients is indicative of a wave pulse encountering a fixed boundary. To explain this,
it is useful to consider the definition of "impedance", which in essence means "resistance". Thus, when
Z2 >>
Z1 , the second material is highly resistive to wave pulse transmission and it effectively acts as a rigid
boundary opposing transmission, so very little of the pulse's energy is successfully transmitted. At the
opposite limit, the wave pulse's energy is easily transmitted into the second medium and the reflection
and transmission coefficients approach those of a free end boundary.
56
Reflection and Transmission Coefficient Variation with Changes in Impedance Ratio
2.5
2
1.5
c
0
U
E
0.5
4U
-0
--
0
60
40
20
80
100
-0.5
-1
-1.5
Z1/Z2
-
Reflection Coefficient
-
Transmission Coefficient
Figure 31: Reflection and Transmission Coefficient Variation with Changes in Impedance Ratio.
57
........
.....
120
3.5 Using the Characteristic Impedance to Characterize an Intermediate Region
Perhaps more relevant to the discussion of ultrasonic NDE applications is the result observed
when a region of intermediate material exists at the interior of a known sample. In particular, NDE can be
utilized to gain information about the nature of the intermediate material in hopes of classifying and
identifying it. The primary sources of information about the unknown material at the interior of a sample
are generally the reflection and transmission coefficients.
Consider the sample shown in Figure 32:
Figure 32: Composite rod with intermediate region (in pulse-echo mode setup).
A wave pulse incident upon this test piece will undergo a number of interactions. First, the wave will
experience both reflection and transmission at the initial boundary, x=xl. At this point, part of the wave's
energy will be reflected back to the point of origin (in this case represented by a transceiver that is able
to interpret and display the signal received) and the remainder will be transmitted into the second,
unknown material. Similarly, upon travelling the extent of the intermediate region, the transmitted
portion of the wave will once again experience the same interactions: part of the energy will be reflected
back towards the point of origin and the remainder will be transmitted into the second region of known
material. This reflected portion will again encounter the first boundary and undergo reflection and
transmission. In this case, the transmitted portion will travel through the initial medium to the transceiver.
It should also be apparent that, in theory, a portion of the wave's energy, however small, will undergo
infinite reflections and remain trapped in the intermediate region. Meanwhile, the portion of the original,
incident wave pulse that is fully transmitted through the intermediate region will continue to travel to the
far end of the sample piece, where it could potentially be received by another device capable of
58
interpreting and displaying the signal. These various interactions are illustrated below. The sample display
in Figure 33, and in Figures 34 - 35 below, show subsequent signals with continually decreasing
amplitudes. This is a direct result of the repeated application of the reflection coefficient (Eq 12), which is
always less than one. Physically, this is indicative of the portions of the incident waveform energy that is
removed from the final waveform(s) received at the transceiver with each additional boundary
interaction. This phenomenon is also highlighted in the various plots included in Appendix D of wave
reflections and transmissions at an interface between selected material pairs.
INCIDENT WAVE
....
.....
.
FIRST TRANSMISSION
SECOND TRANSMISSION
REFLECTION
SECOND
FIRST REFLECTION
Sample Display
CL
C:
I
A
/\
t
Later Signals
Initial
Pulse
1t Reflection
Figure 33: Boundary interactions with sample display.
In order to fully describe the total effect of this sequence of interface interactions, each
interaction will be addressed individually and the resulting mathematical expressions stated or derived.
For clarity, all material properties associated with the initial and final material regions (since these two
regions contain identical material in the schematic) will be denoted with the subscript 'A'. Material
59
properties of the interior region of "unknown" material will be designated with the subscript 'B'.
Reflection and transmission coefficients will be identified by the material of propagation of the wave
function that is the incident wave for each individual boundary interaction. The additional subscripts ',
'r', and 't', previously used to identify the incident, reflected, and transmitted waves, respectively, will
again be utilized here.
The first boundary interaction is identical to the interaction previously discussed, where an
incident wave pulse travels through a rod and encounters an internal interface within the rod between its
original medium of propagation and another, dissimilar medium. As such, the waveforms resulting from
this interaction are given by Eqs 10 and 11. At this point, the reflected wave travels back to the transceiver
and is the first signal received and observed by the administrator. Since this wave travels entirely in a
known medium, its speed for its entire journey is also known. As such, the location of the first interface,
xi, can be obtained directly, using the length of time that elapses between initial signal generation and
the observation of the reflected wave.
60
E A1 ,c,
E2 ,A 2 ,
C2
x
X1
X2
Sample Display
E-
E
M1
t
14 Reflection
Initial
Pulse
Figure 34: Sample calculation offirst boundary in composite rod.
The reflection coefficient contains information about the material properties of the intermediate
region of unknown material. Eq 23 gives an expression for the characteristic impedance of the unknown
material in terms of the reflection coefficient, which can be directly measured after the first reflection is
observed.
R
A
= Reflected Amplitude _ ZA-ZB -nd>ZB Api1-RtdA ZA (Eq 23)
Incident Amplitude
ZA+ZB
-B =
ZA
1-R
(A
\
+ RTA
(Eq
24)
G+RfA
The second signal observed will be the portion of the reflection at the second boundary that is
transmitted back into the original material, and the third signal will be the portion of the original wave
pulse that is transmitted completely through the rod and back to the point of origin without any
reflections at any of the internal interfaces. The elapsed time between initial wave pulse generation and
61
the receipt of the various subsequent signals is used to obtain an expression for the wave speed in the
unknown medium which, in turn, can be used to locate the second internal interface and reveal the extent
of the intermediate region of unknown material.
The time delay to the receipt of the second signal is given by:
At 2
= At1 + 2
x2-X)
\
CB
/
= 2
(LI +
X2-X1)
CA
(Eq 26)
2CA(x 2 -x 1 )
CB
(Eq 25)
CB
Ct2CA-2x,
The unknown quantities of Eq 26 are highlighted in red." Naturally, the presence of two
unknowns predicates the need for a second equation to fully solve the system. Thus, the time required
for the fully transmitted pulse to traverse the entire test piece is given by:
CB
L+
2 (E
CA
X2-X1
CB
+
2CA(x2-x 1 )
CAt3-2xl-2(X 3 -X
X3-X2
CA
(Eq 27)
(Eq 28)
2
)
At 3 =
Eq 26 and Eq 28 complete a simple system of two equations that can be solved for X2 by eliminating the
unknown primary wave speed, cB. Thus, the location of the second interface between the two materials
can be found.
12
X2 =
C.(At
2
- At 3 )
+ 2x 3
(Eq 29)
Now that the extent of the intermediate region is known and the unknown material has been
identified by its characteristic impedance, the problem is fully described.
11 Note that if the characteristic impedance provided sufficient information to explicitly identify the unknown
material, perhaps through the use of the plots in Appendix B, then the wave speed of the unknown medium,
known as well and X2 can be calculated directly.
12 This description assumes that the total length of the rod, x3, is known.
62
CB,
is
O~D
'K
I
I
Initial
Pulse
3 rd
2 " Reflection
1st Reflection
R eflection
Sample Display
Initial
Pulse
A
ft Reflection
V/
CJ
E
2 nd
Reflection
3 rd
Reflection
\/
<- Ati
<
<
\/
A
>
t
At 2
At3
>
Figure 35: Sample display showing elapsed time intervals between receipt of consecutive signals.
63
In an alternative test arrangement where the far end of the test piece is accessible and provides
information (i.e. through-transmission mode analysis), the equations are slightly modified. Consider the
case where the signal generator at the originating end only generates a wave form and is not capable of
receiving/ interpreting any signals, as shown in Figure 12 of Section 2.3.
If only the far end of the test piece receives output, the first signal received will be the wave pulse
that is fully transmitted directly through the specimen with no internal reflections. As such, its wave
function will only contain the transmission coefficients between the two dissimilar materials.
ft(Xt - c 2 t)
T
=
BAfi(xi
-
(Eq 30)
c 1 t)
Again, the ratio between the received wave form and the initial incident wave form can be used to
determine the characteristic impedance of the unknown material.
Transmitted Wave
_
OriginalIncident Wave =
=_
B
fA
ZA
ZB
1
4ZA
2ZB
\ZB+ZA
2ZA
ZB+ZA
(Eq
31)
(Eq 32)
TB_ B T1
-
ZA
4z
TBTA-1
(Eq 33)
As before, the time lapses between consecutive signals will be used to determine the extent of
the intermediate region. However, without the benefit of any signal that only travels in the known
medium and never passes through the intermediate region (recall the first signal observed by the
transceiver in the previous example), the location of the first boundary cannot be found directly and
becomes another unknown in the eventual system of equations, thus requiring a third equation in order
to solve the system completely.
The most obvious choice for the first equation comes from the time interval between the release
of the initial incident wave form and the receipt of the fully transmitted wave, which is just one-half of
the time interval already found in Eq 27. For the remaining two equations, any number of equations can
be developed.
13
For simplification of calculations, the second equation will be defined by the path of the
wave pulse that reflects twice in the first region of known material and is then transmitted the rest of the
way through the test piece. The final equation will be defined by the path of the wave pulse that reflects
It is prudent to utilize information from the earliest signals observed since signal strength will diminish with
successive signals and later signals may become harder to detect and observe. This is why the example provided
develops the equations for the first few signals that will be observed. Also, the pulses that experience internal
reflections in the known medium are chosen in order to capitalize on as much a priori knowledge as possible, namely
the wave speed of the known material making up the first and last regions of propagation.
13
64
back into the known medium at the far end, reflects again at the second boundary of the intermediate
region, and is intercepted again at the far end.
Initial
T,
Pulse
T2>
_
Initial
/Ri
N
T2
T1
Initial
Pulse
R1
T
Figure 36: Wave pulse paths (through-transmission mode calculation).
Thus the complete system of equations used to determine the locations of the two internal interfaces is:
(Eq
34)
Ati =L+X2-X1 +X3-X2
At 2 =
3x 1
At 3 =
CA
CA
-
CA
CB
CA
+
+
X2
X1
CB
X2-X1
CB
+
X3 X
CA
2
+ 3(x 3-x 2 )
(Eq 35)
(Eq 36)
CA
Note that the subscripts of the last two time intervals are arbitrary and do not necessarily indicate the
order of the signals observed. While the first signal, which undergoes no internal reflections, will always
be the first wave form to reach the far end of the test piece, determining which of these last two signals
will reach the far end first depends entirely on the location of the intermediate region. More precisely, it
65
depends on which of the two regions of the known material presents the longer travel path for the wave
4
forms. If the two regions are equal in their length, At 2 = At 3.1
Initial
)
Pulse
RI
T2 \
R2K
Ti
Pulse 2 is receive-d first.
MMW*.
Initial
Pulse
\
R
T2
T)
\
(
R2 4
Pulse 3 is received first.
Figure 37: Comparison of through-transmission mode path lengths.
Solving this system yields equations for the locations of the first and second boundaries in terms of the
(known) wave speed of the initial medium of propagation and the (directly measured) time intervals
between the receipt of the various wave pulses, thus describing the physical extent of the intermediate
region.
x1
=
2
X2= X3-
(At2 -
LA (At3
2
(Eq 37)
At,)
-
At,)
(Eq 38)
3.6 Using the Characteristic Impedance to Identify an Unknown Material at an Interface
Using the material discussed and derived in the previous section, characterizing an unknown
material that shares an interface with a known material (as shown) is quite straightforward.
In the event that two regions are equal, Eq 35 and Eq 36 are equivalent and the system is reduced to just two
equations. However, there will only be two unknowns because finding either x, or x2 explicitly automatically defines
the location of the other interface, assuming the total rod length is known.
14
66
Figure 38: Composite rod with single known material/ unknown material interface.
Assuming that the reflection or transmission coefficient can be found, using the same testing
setups already described in previous sections, Eq 23 or Eq 32, in conjunction with the plots found in
Appendix B, provide the means to characterize the unknown material beyond the boundary, assuming the
materials present in the composite rod are included in the Table of Material Properties in Appendix A.
Since it is assumed that the unknown region carries on "indefinitely", there is no requirement to solve for
the physical extent of the region of unknown material. This might be the case if the unknown material
carries on through the test piece to such an extent as to be approximated as "infinite", or any other
scenario when only the identity of the unknown material is desired but not its physical dimensions.
In the event that both materials in a test piece are unknown, the table in Appendix B could provide
adequate information for material characterization if both of the materials are common materials that
are included in the table. The expectation is that the test administrator would measure the reflection or
transmission coefficient, as before, and use it to obtain an expression for the ratio of the characteristic
impedances of the two unknown materials (Eqs 24 and 33). Naturally, it is all but impossible to create
such a table for every material, alloy, and composite in existence, so only selected common materials have
been included.
4. Conclusion
One primary benefit of ultrasonic NDE as an evaluation technique is that it acts as a volumetric
NDE process capable of yielding information about discontinuities within the body of a sample piece. The
derivations of the mathematical expressions necessary to carry out such an analysis on the simplified case
of a composite rod are presented in the main body and/or appendices of this thesis.
4.1 Derivation Methodology
The derivations of the reflection and transmission coefficients, which represent the central
theme of this thesis, are presented in detail in Appendix E. The coefficients are derived from basic
principles of mechanical wave behavior, as discussed in the introductory portions of the main thesis body.
67
In particular, the principal wave speed, which is defined for mechanical waves in terms of the physical
properties of the medium of propagation (namely, the density and modulus of elasticity), and the principle
of superposition, along with applicable conditions of continuity at the interface, combine to yield
expressions for the reflected and transmitted waves at an interface in terms of the incident waveform.
With regard to the mathematical techniques applied, the level of difficulty is minimal. A small
amount of elementary calculus is supplemented by some algebraic manipulations, and the language
should be understandable to the majority of readers. Additional information about waveform phenomena
and wave dynamics, provided in the supplemental references, may be of interest to some readers due to
the limited scope of this thesis. In particular, the discussions of this thesis are limited to non-dispersive
media, which is a valid assumption for the purposes of ultrasonic NDE due to the limited range of typical
wave propagations in NDE processes.
4.2 Material Characterization
A proposed procedure for using the reflection and transmission coefficients to identify a foreign
material present at the interior of a long, thin rod is outlined. The main procedure discussed utilizes the
definition of characteristic impedance, and material plots are provided in Appendix B that can be applied
to material characterization in an ultrasonic NDE process. The timing of receipt of subsequent signals can
also be used to characterize the extent of the foreign material present at the interior of a test piece.
An interesting result of targeted analysis of the behaviors of the reflection and transmission
coefficients under various conditions of comparisons between the properties of adjacent segments in a
composite rod was the observation of equivalent behaviors. These equivalencies indicate that the
behavior of a waveform at an interface between somewhat special material pairs, where either the
principal wave speed or the modulus of elasticity is very similar between the two materials, can be
mimicked with a rod comprised of only one material and an appropriate differential between segment
cross-sectional areas. Applications of this observation might include an instance when it is desirable to
review the interactions at an interface of a particular material pair but only one material is available.
4.3 Areas for Further Study
Since the scope of this thesis is limited to a one-dimensional test piece, it might be beneficial to
expand all relevant equations to three dimensions to allow for solutions that are applicable to a generic
test piece rather than just long, thin rods. One might also suggest expanding the discussion to include
dispersive media but, as has been previously emphasized, the limited spatial extent of a typical NDE
process would limit the effects of dispersion enough as to render them negligible.
68
Appendix A: Table of Material Properties
The following is a compilation of material properties that are relevant to the discussion of this thesis. Included in the table is a small sampling of
solids commonly used across a variety of industries. As with all material properties, the values presented in the table are approximate because
measured values fluctuate based on instrument accuracy, sample purity, and measurement conditions, such as ambient temperature.
Boron/ E poxy3
195.
4072.9
6.3
131.6
2000.0
3.88
9.87
32.40
1.7
5.82
72.0
1503.8
31.3
653.7
2203.0
4.27
5.72
18.76
3.77
12.37
199.9
4176.0
75.8
1584.0
875.5
16.06
4.91
16.12
3.1
9.98
Cast Iron
Alloys 2
Glass (Fused
Metrylte
Silica)S
Graphite!
Epoxy26,
Nic:Stel"
Boron/
1Modulus of Elasticity and Shear Modulus of Elasticity obtained from: http://www.engineersedge.com/manufacturing spec/average properties structural_materials.htm
2Density
obtained from: http://www.engineeringtoolbox.com/density-solids-d_1265.html and http://www.engineeringtoolbox.com/density-materials-d_1652.html
69
3 Material properties obtained from: http://www.matweb.com/search/DataSheet.aspx; Information provided by Specialty Materials Inc. Values specified refer to Specialty
Materials' 5521, which is
a highly advanced pre-impregnated tape that combines Specialty Materials' 4.0 or 5.6 mil boron filament with a tough, rigid, epoxy resin.
4 Modulus of Elasticity and Shear Modulus of Elasticity obtained from: Metal Material Data Sheets. Online version available at:
maximum values selected for inclusion in table
http://app.knoveI.com/hotlink/toc/id:kpMMDS0002/meta1-materia1-data-sheets/meta1-materiaI-data-sheets;
s Material properties of glass obtained from: http://en.wikipedia.org/wiki/Listof physical_propertiesofglass
6 Modulus of Elasticity obtained from: Airframe Structural Design - Practical Design Information and Data on Aircraft Structures (2nd Edition)
7 Shear Modulus of Elasticity obtained from: Carbon / Epoxy Composite Materials - Properties - Supplier Data by Goodfellow. Value specified refers to carbon/epoxy composite
sheet.
obtained from: ASM Handbook, Volume 18 - Friction, Lubrication, and Wear Technology; Density range arises from variations in nickel content of compound. Density
rises with increase of
nickel content.
9 Material properties obtained from: http://www.matbase.com/material-categories/natural-and-synthetic-polymers/
8 Density
70
Appendix B: Material Characterization using Characteristic Impedance:
Table and Plots
Intended to aid in the characterization of materials in an ultrasonic NDE test of a composite rod,
the single table and various plots included in this appendix provide information on the following common
materials: aluminum, beryllium copper, boron epoxy, cast iron, glass, graphite epoxy, nickel steel,
polymethyl methacrylate (PMMA), stainless steel, and titanium.1 5 To use the plots for material
characterization, first calculate the characteristic impedance ratio of the two dissimilar materials in the
composite test piece (Materials A and B in the schematic below) using either the reflection or the
transmission coefficient, depending on testing method/ mode.
Pulse - Echo Mode:
ZB
ZA
Through -Transmission Mode:
ZB
ZA
=
::
-R
G+RfA)
41
4ZA
1
Note that in pulse-echo mode, the reflection coefficient, R , can be found experimentally as the
ratio of the amplitudes of the first signal received at the transceiver and the generated incident wave
pulse. In through-transmission mode, the product T B TA will be equal to the ratio of the amplitudes of
the first signal received and the generated incident wave pulse.
When the original propagating material (Material A in the schematics) is known, refer to that
material's plots and use either the reflection or the transmission coefficient, as obtained through
ultrasonic testing, to determine the second material in the test piece. The plots provide valid material
characterization information for either of the test piece schematics shown below. In the event that the
initial propagating material is unknown, test administrators can use the included table to attempt to
characterize the two materials present in the test piece from the characteristic impedance ratio obtained
above.
Material A
Material A
Material
Material
Material A
A'
Figure B 1: Valid schematics for use of material characterization table/plots.
15
The table and plots were developed utilizing the data provided in Appendix A: Table of Material Properties.
71
Material Characterization Table for Unknown Propagating Material of Incident Waveform
0- CLx
_
x
W~
00~
0
CL
Aluminum Alloys
1
2.282
1.415
2.999
0.902
1.629
2.914
0.138
2.750
1.621
Beryllium Copper
0.438
1
0.620
1.314
0.395
0.714
1.277
0.061
1.205
0.710
Boron/ Epoxy
Cast Iron Alloys
Glass (Fused Silica)
Graphite/ Epoxy
Nickel Steel
Polymethyl Methacrylate
(PMMA)
Stainless Steel
Titanium
0.707
0.333
1.108
0.614
0.343
1.613
0.761
2.529
1.401
0.783
1
0.472
1.568
0.869
0.486
2.120
1
3.323
1.841
1.029
0.638
0.301
1
0.554
0.310
1.151
0.543
1.805
1
0.559
2.060
0.972
3.229
1.789
1
0.098
0.046
0.153
0.085
0.047
1.943
0.917
3.047
1.688
0.944
1.146
0.541
1.797
0.995
0.556
7.236
16.514
10.239
21.702
6.530
11.789
21.088
1
19.898
11.733
0.364
0.617
0.830
1.408
0.515
0.873
1.091
1.850
0.328
0.557
0.592
1.005
1.060
1.797
0.050
0.085
1
1.696
0.590
1
72
Reflection Coefficients of Interface from Generic Aluminum Alloy to Various Common Materials
1.000
PMMA
0.800
e
0.600
0.400
0
U
C
0
0.200
Glass (Fused Silica)
0
0.000
0.00
0.50
1.00
3.00
2.50
2.00
1.50
Titanium
-0.200
Boron/ Epoxy
Graphite/ Epoxy
Beryllium Copper
-0.400
Stainless Steel
Cast Iron Alloys
Nickel Steel
-0.600
Characteristic Impedance Ratio
73
.
...............
-d
..........
....
....
.................
(Z/ZALUMINUM)
3.50
Transmission Coefficients of Interface from Generic Aluminum Alloy to Various Common Materials
2.000
PMMA
1.800
1.600
1.400
Glass (Fused Silica)
1.200
0
a
0
1.000
Boron/ Epoxy
Titanium
E
0.800
Beryllium Copper
Graphite/ Epoxy
0.600
Nickel Steel
Stainless Steel
Cast Iron Alloys
0.400
0.200
0.000
0.00
0.50
1.00
2.00
1.50
Characteristic Impedance Ratio (Z/ZALUMINUM)
74
2.50
3.00
3.50
Reflection Coefficients of Interfacefrom Beryllium Copper to Various Common Materials
1.000
* PMMA
0.800
0.600
U
0
U
0
*
Glass (Fused Silica)
0.400
* Aluminum Alloys
U
* Boron/ Epoxy
0.200
Titanium
Graphite/ Epoxy
0.000
0.000
0.200
0.400
0.800
0.600
1.000
1.200
1.400
Nickel Steel
Stainless Steel
-0.200
Cast Iron Alloys
Impedance Ratio
75
(Z/ZBERYLLIUM cOPPER)
Transmission Coefficients of Interface from Beryllium Copper to Various Common Materials
2.000
0 PMMA
1.800
1.600
Glass (Fused Silica)
1.400
Aluminum Alloys
Boron/ Epoxy
rpTitanium
A) 1.200
C
Graphite/ Epoxy
1.000
Stainless Steel
0.800
Nickel Steel
Cast Iron Alloys
0.600
0.400
0.200
0.000
-
0
U
C
0
0.000
0.200
0.400
0.800
0.600
Impedance Ratio
76
(Z/ZBERYLLIUM COPPER)
1.000
1.200
1.400
Reflection Coefficients of Interfacefrom Boron Epoxy to Various Common Materials
1.000
0.800
0
PMMA
0.600
0.400
Glass (Fused Silica)
0)
U
C
0
0.200
Aluminum Alloys
a)
Titanium
0.000
0.600
0.500
2.500
2.000
1.500
1.000
Beryllium Copper
Graphite/ Epoxy
-0.200
rStainless
Steel
Cast Iron Alloys
-0.400
-0.600
Nickel Steel
Impedance Ratio (Z/ZBORON
77
. ............................
............................................
EPOXY)
Transmission Coefficients of Interfacefrom Boron Epoxy to Various Common Materials
2.000
1.800
0 PMMA
1.600
1.400
Glass (Fused Silica)
w 1.200
Aluminum Alloys
U
C
0
Titanium
1.000
Beryllium Copper
E
Graphite/ Epoxy
0.800
Stainless Steel
0.600
Nickel Steel
Cast Iron Alloys
0.400
0.200
0.000
0.000
0.500
1.500
1.000
Impedance Ratio (Z/ZBORON
78
.......
............
EPOXY)
2.000
2.500
Reflection Coefficients of Interfacefrom Generic Cast Iron Alloy to Various Common Materials
1.000
0.900
* PMMA
0.800
0.700
0.600
Glass (Fused Silica)
0
U
C
0.500
0
Aluminum Alloys
Boron/ Epoxy
S0.400
Titanium
0.300
Graphite/ Epoxy
0.200
* Beryllium Copper
0.100
* Stainless Steel
0.000
0.000
* Nickel Steel
0.200
0.800
0.600
0.400
Impedance Ratio (Z/ZCAST
79
IRON)
1.000
1.200
Transmission Coefficients of Interface from Generic Cast Iron Alloy to Various Common Materials
2.500
2.000
0 PMMA
Glass (Fused Silica)
Boron/ Epoxy
1.500
/
/
Titanium
Aluminum Alloys
r0
U
Beryllium Copper
V1
0
Stainless Steel
Graphite/ Epoxy
E
Lin
C
L-
1.000
Nickel Steel
0.500
0.000
L
0.000
0.200
0.600
0.400
Impedance Ratio (Z/ZcAsT IRON)
80
0.800
1.000
1.200
Reflection Coefficients of Interface from Glass to Various Common Materials
0.800
* PMMA
0.600
0.400
0.200
Aluminum Alloys
0
4-j
U
0.000
a>
0.000
0.500
1.000
2.000
1.500
3.000
2.500
3.500
C
Boron/ Epoxy
-0.200
Graphite/ Epoxy
Titanium
Beryllium Copper
-0.400
Stainless Steel
-0.600
-
Nickel Steel
Cast Iron Alloys
-0.800
Impedance Ratio (Z/ZGLASS)
81
Transmission Coefficients of Interfacefrom Glass to Various Common Materials
2.000
1.800
PMMA
1.600
1.400
.2 1.200
Aluminum Alloys
0
U
1.000
Boron/ Epoxy
E
Graphite/ Epoxy
0.800
Stainless Steel
0.600
Titanium
0 Beryllium Copper
7r
Cast Iron Alloys
0.400
Nickel Steel
0.200
0.000
0.000
0.500
1.000
2.000
1.500
Impedance Ratio (Z/ZGLASS)
82
2.500
3.000
3.500
Reflection Coefficients of Interfacefrom Graphite Epoxy to Various Common Materials
1.000
PMMA
0.800
0.600
0.400
Glass (Fused Silica)
C
Aluminum Alloys
0
0.200
Boron/ Epoxy
Titanium
0.000
0.000
0.200
0.400
0.600
1.000
0.800
1.200
1.400
1.600
1.800
2.000
Beryllium Copper
Nickel Steel
-0.200
Stainless Steel
-0.400
Impedance Ratio
83
(Z/ZGRAPHITE)
Cast Iron Alloys
Transmission Coefficients of Interface from Graphite Epoxy to Various Common Materials
2.000
PMMA
0
1.800
1.600
Glass (Fused Silica)
1.400
Aluminum Alloys
.~1.200
Boron/ Epoxy
Titanium
0
U
C1.000
Beryllium Copper
Stainless Steel
Nickel Steel
Cast Iron Alloys
S0.800
0.600
0.400
0.200
0.000
0.000
-
0
0.200
0.400
0.600
1.000
0.800
Impedance Ratio
84
1.200
(Z/ZGRAPHITE)
1.400
1.600
1.800
2.000
Reflection Coefficients of Interface from Nickel Steel to Various Common Materials
1.000
PMMA
0
0.800
0.600
Glass (Fused Silica)
Aluminum Alloys
0
U
C
0
Boron/ Epoxy
0.400
Graphite/ Epoxy
U
Titanium
0.200
Beryllium Copper
Stainless Steel
0.000
0.000
-0.200
0.200
0.600
0.400
Impedance Ratio
85
(Z/ZNICKEL STEEL)
0.800
Cast Iron Alloys
1.000
1.200
Transmission Coefficients of Interface from Nickel Steel to Various Common Materials
2.500
2.000
PMMA
Glass (Fused Silica)
Aluminum Alloys
Boron/ Epoxy
U
1.500
Graphite/ Epoxy
4
0
U
Beryllium Copper
C
0
Titanium
E
1.000
0.500
0.000
0.000
-
C
I-
Stainless Steel
Cast Iron Alloys
Ln
0.200
0.800
0.600
0.400
Impedance Ratio
1.200
1.000
(Z/ZNICKEL STEEL)
86
............
..
.
.....
... ......
Reflection Coefficients of Interfacefrom Poly (methyl methacrylate) to Various Common Materials
0.000
0.000
10.000
5.000
15.000
20.000
25.000
-0.100
-0.200
-0.300
-0.400
cu
0
oC -0.500
0
U
cu
w -0.600
Glass (Fused Silica)
Aluminum Alloys
-0.700
Beryllium Copper
Graphite/ Epoxy
Stainless Steel
- Titanium
-0.800
Cast Iron Alloys
Boron/ Epoxy
-0.900
Nickel Steel
-1.000
Impedance Ratio (Z/ZpmMA)
87
Transmission Coefficients of Interface from Poly (methyl methacrylate) to Various Common Materials
0.300
Glass (Fused Silica)
Aluminum Alloys
0.250
0.200
Boron/ Epoxy
Titanium
0
U
-)
C0.150
Graphite/ Epoxy
Beryllium Copper
E
Stainless Steel
Nickel Steel
Cast Iron Alloys
0.100
0.000
-
0.050
0.000
5.000
15.000
10.000
Impedance Ratio
88
(Z/ZPMMA)
20.000
25.000
Reflection Coefficients of Interfacefrom Stainless Steel to Various Common Materials
1.000
PMMA
0.800
0.600
Glass (Fused Silica)
Aluminum Alloys
a
U
C
0
Boron/ Epoxy
0.400
Titanium
0.200
Graphite/ Epoxy
Beryllium Copper
Nickel Steel
Cast Iron Alloys
0.000
0.00
-0.200
0.200
0.600
0.400
Impedance Ratio
89
(Z/ZsTAINLEss STEEL)
0.800
1.000
1.200
Transmission Coefficients of Interface from Stainless Steel to Various Common Materials
PMMA
2.000
1.800
Glass (Fused Silica)
1.600
I
Boron/ Epoxy
Titanium
1.400
Aluminum AlloysI
Beryllium Copper
1.200
Graphite/ Epoxy
0
U
Nickel Steel
C 1.000
.2
E
Cast Iron Alloys
0.800
0.600
0.400
0.200
0.000
0.000
0.600
0.400
0.200
Impedance Ratio
90
RPM_
.
................
(Z/ZSTAINLESS STEEL)
0.800
1.000
1.200
Reflection Coefficients of Interfacefrom Titanium to Various Common Materials
1.000
s
PMMA
0.800
0.600
0.400
Glass (Fused Silica)
0
4-.J
0.200
Aluminum Alloys
Boron/ Epoxy
Graphite/ Epoxy
-
0.000
0.000
0.200
0.400
0.600
1.000
0.800
1.200
1.400
1.600
Stainless Steel
-0.200
Beryllium Copper
Nickel Steel -'
-0.400
Impedance Ratio
91
(Z/ZTITANIUM)
1.800
2.000
Cast Iron Alloys
Transmission Coefficients of Interfacefrom Titanium to Various Common Materials
2.000
PMMA
1.800
1.600
Glass (Fused Silica)
1.400
Boron/ Epoxy
1.200
Graphite/ Epoxy
Aluminum Alloys
0
U
C
1.000
0
-
Beryllium Copper
E
Stainless Steel
Cast Iron Alloys
u0.800
Nickel Steel
0.600
0.400
0.200
0.000
0.000
0.200
0.400
0.600
1.000
0.800
Impedance Ratio
92
.....
1.200
(Z/ZTITANIUM)
1.400
1.600
1.800
2.000
Appendix C: Tabulated Reflection and Transmission Coefficients for Pairings of Selected Common Materials
Aluminum Alloys
Aluminum Alloys
Aluminum Alloys
Aluminum Alloys
Aluminum Alloys
Aluminum Alloys
Aluminum Alloys
Aluminum Alloys
Aluminum Alloys
Beryllium Copper
Beryllium Copper
Beryllium Copper
Beryllium Copper
Beryllium Copper
Beryllium Copper
Beryllium Copper
Beryllium Copper
Beryllium Copper
Boron/ Epoxy
Boron/ Epoxy
Boron/ Epoxy
Boron/ Epoxy
Boron/ Epoxy
Boron/ Epoxy
Boron/ Epoxy
Boron/ Epoxy
Boron/ Epoxy
Cast Iron Alloys
Cast Iron Alloys
Cast Iron Alloys
Cast Iron Alloys
Cast Iron Alloys
Cast Iron Alloys
Cast Iron Alloys
Cast Iron Alloys
Cast Iron Alloys
Glass (Fused Silica)
Glass (Fused Silica)
Glass (Fused Silica)
Glass (Fused Silica)
Glass (Fused Silica)
Glass (Fused Silica)
Glass (Fused Silica)
Glass (Fused Silica)
Glass (Fused Silica)
Graphite/ Epoxy
Graphite/ Epoxy
Graphite/ Epoxy
Graphite/ Epoxy
Graphite/ Epoxy
Graphite/ Epoxy
Graphite/ Epoxy
Graphite/ Epoxy
Graphite/ Epoxy
Nickel Steel
Nickel Steel
Nickel Steel
Nickel Steel
Nickel Steel
Nickel Steel
Nickel Steel
Nickel Steel
Nickel Steel
Beryllium Copper
Boron/ Epoxy
Cast Iron Alloys
Glass (Fused Silica)
Graphite/ Epoxy
Nickel Steel
Polymethyl Methacrylate (PMMA)
Stainless Steel
Titanium
Aluminum Alloys
Boron/ Epoxy
Cast Iron Alloys
Glass (Fused Silica)
Graphite/ Epoxy
Nickel Steel
Polymethyl Methacrylate (PMMA)
Stainless Steel
Titanium
Aluminum Alloys
Beryllium Copper
Cast Iron Alloys
Glass (Fused Silica)
Graphite/ Epoxy
Nickel Steel
Polymethyl Methacrylate (PMMA)
Stainless Steel
Titanium
Aluminum Alloys
Beryllium Copper
Boron/ Epoxy
Glass (Fused Silica)
Graphite/ Epoxy
Nickel Steel
Polymethyl Methacrylate (PMMA)
Stainless Steel
Titanium
Aluminum Alloys
Beryllium Copper
Boron/ Epoxy
Cast Iron Alloys
Graphite/ Epoxy
Nickel Steel
Polymethyl Methacrylate (PMMA)
Stainless Steel
Titanium
Aluminum Alloys
Beryllium Copper
Boron/ Epoxy
Cast Iron Alloys
Glass (Fused Silica)
Nickel Steel
Polymethyl Methacrylate (PMMA)
Stainless Steel
Titanium
Aluminum Alloys
Beryllium Copper
Boron/ Epoxy
Cast Iron Alloys
Glass (Fused Silica)
Graphite/ Epoxy
Polymethyl Methacrylate (PMMA)
Stainless Steel
Titanium
5.04
5.04
5.04
5.04
5.04
5.04
5.04
5.04
5.04
3.90
3.90
3.90
3.90
3.90
3.90
3.90
3.90
3.90
9.87
9.87
9.87
9.87
9.87
9.87
9.87
9.87
9.87
5.73
5.73
5.73
5.73
5.73
5.73
5.73
5.73
5.73
5.72
5.72
5.72
5.72
5.72
5.72
5.72
5.72
5.72
9.10
9.10
9.10
9.10
9.10
9.10
9.10
9.10
9.10
4.91
4.91
4.91
4.91
4.91
4.91
4.91
4.91
4.91
3.90
9.87
5.73
5.72
9.10
4.91
2.52
4.96
5.03
5.04
9.87
5.73
5.72
9.10
4.91
2.52
4.96
5.03
5.04
3.90
5.73
5.72
9.10
4.91
2.52
4.96
5.03
5.04
3.90
9.87
5.72
9.10
4.91
2.52
4.96
5.03
5.04
3.90
9.87
5.73
9.10
4.91
2.52
4.96
5.03
5.04
3.90
9.87
5.73
5.72
4.91
2.52
4.96
5.03
5.04
3.90
9.87
5.73
5.72
9.10
2.52
4.96
5.03
70.3
70.3
70.3
70.3
70.3
70.3
70.3
70.3
70.3
124.1
124.1
124.1
124.1
124.1
124.1
124.1
124.1
124.1
195.0
195.0
195.0
195.0
195.0
195.0
195.0
195.0
195.0
240.0
240.0
240.0
240.0
240.0
240.0
240.0
240.0
240.0
72.0
72.0
72.0
72.0
72.0
72.0
72.0
72.0
72.0
206.8
206.8
206.8
206.8
206.8
206.8
206.8
206.8
206.8
199.9
199.9
199.9
199.9
199.9
199.9
199.9
199.9
199.9
Grouped by Ratio of Segment Cross-Sectional Areas
124.1
195.0
240.0
72.0
206.8
199.9
7.6
190.3
113.8
70.3
195.0
240.0
72.0
206.8
199.9
7.6
190.3
113.8
70.3
124.1
240.0
72.0
206.8
199.9
7.6
190.3
113.8
70.3
124.1
195.0
72.0
206.8
199.9
7.6
190.3
113.8
70.3
124.1
195.0
240.0
206.8
199.9
7.6
190.3
113.8
70.3
124.1
195.0
240.0
72.0
199.9
7.6
190.3
113.8
70.3
124.1
195.0
240.0
72.0
206.8
7.6
190.3
113.8
-0.916
-0.868
-0.936
-0.801
-0.884
-0.934
-0.368
-0.930
-0.884
-0.629
-0.722
-0.859
-0.597
-0.754
-0.855
0.026
-0.847
-0.753
-0.752
-0.883
-0.910
-0.729
-0.840
-0.907
-0.209
-0.902
-0.839
-0.538
-0.768
-0.650
-0.501
-0.689
-0.813
0.162
-0.803
-0.688
-0.835
-0.924
-0.880
-0.942
-0.895
-0.940
-0.412
-0.936
-0.895
-0.720
-0.867
-0.794
-0.897
-0.694
-0.894
-0.141
-0.888
-0.817
-0.548
-0.773
-0.658
-0.823
-0.511
-0.696
0.148
-0.808
-0.695
0.084
0.132
0.064
0.199
0.116
0.066
0.632
0.070
0.116
0,371
0.278
0.141
0.403
0.246
0.145
1.026
0.153
0.247
0.248
0.117
0.090
0.271
0.160
0.093
0.791
0.098
0.161
0.462
0.232
0.3S0
0.499
0.311
0.187
1.162
0.197
0.312
0.165
0.076
0.120
0.058
0.105
0.060
0.588
0.064
0.105
0.280
0.133
0.206
0.103
0.306
0.106
0.859
0.112
0.183
0.452
0.227
0.342
0.177
0.489
0.304
1.148
0.192
0.305
93
Tabulated Reflection and Transmission Coefficients
Polymethyl Methacrylate (PMMA)
Polymethyl Methacrylate (PMMA)
Polymethyl Methacrylate (PMMA)
Polymethyl Methacrylate (PMMA)
Polymethyl Methacrylate (PMMA)
Polymethyl Methacrylate (PMMA)
Polymethyl Methacrylate (PMMA)
Polymethyl Methacrylate (PMMA)
Polymethyl Methacrylate (PMMA)
Stainless Steel
Stainless Steel
Stainless Steel
Stainless Steel
Stainless Steel
Stainless Steel
Stainless Steel
Stainless Steel
Stainless Steel
Titanium
Titanium
Titanium
Titanium
Titanium
Titanium
Titanium
Titanium
Titanium
Aluminum Alloys
Beryllium Copper
Boron/ Epoxy
Cast Iron Alloys
Glass (Fused Silica)
Graphite/ Epoxy
Nickel Steel
Stainless Steel
Titanium
Aluminum Alloys
Beryllium Copper
Boron/ Epoxy
Cast Iron Alloys
Glass (Fused Silica)
Graphite/ Epoxy
Nickel Steel
Polymethyl Methacrylate (PMMA)
Titanium
Aluminum Alloys
Beryllium Copper
Boron/ Epoxy
Cast Iron Alloys
Glass (Fused Silica)
Graphite/ Epoxy
Nickel Steel
Polymethyl Methacrylate (PM MA)
Stainless Steel
2.52
2.52
2.52
2.52
2.52
2.52
2.52
2.52
2.52
4.96
4.96
4.96
4.96
4.96
4.96
4.96
4.96
4.96
5.03
5.03
5.03
5.03
5.03
5.03
5.03
5.03
5.03
5.04
3.90
9.87
5.73
5.72
9.10
4.91
4.96
5.03
5.04
3.90
9.87
5.73
5.72
9.10
4.91
2.52
5.03
5.04
3.90
9.87
5.73
5.72
9.10
4.91
2.52
4.96
Grouped by Ratio of Segment
Cross-Sectional Areas
7.6
7.6
7.6
7.6
7.6
7.6
7.6
7.6
7.6
190.3
190.3
190.3
190.3
190.3
190.3
190.3
190.3
190.3
113.8
113.8
113.8
113.8
113.8
113.8
113.8
113.8
113.8
70.3
124.1
195.0
240.0
72.0
206.8
199.9
190.3
113.8
70.3
124.1
195.0
240.0
72.0
206.8
199.9
7.6
113.8
70.3
124.1
195.0
240.0
72.0
206.8
199.9
7.6
190.3
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.1
-0.958
-0.981
-0.970
-0.986
-0.9S3
-0.974
-0.985
-0.984
-0.974
-0.569
-0.785
-0.675
-0.832
-0.533
-0.711
-0.828
0.119
-0.710
-0.721
-0.867
-0.794
-0.897
-0.695
-0.819
-0.895
-0.143
-0.889
0.042
0.019
0.030
0.014
0.047
0.026
0.015
0.016
0.026
0.431
0.215
0.325
0.168
0.467
0.289
0.172
1.119
0.290
0.279
0.133
0.206
0.103
0.305
0.181
0.105
0.857
0.111
94
Tabulated Reflection and Transmission Coefficients
Aluminum Alloys
Aluminum Alloys
Aluminum Alloys
Aluminum Alloys
Aluminum Alloys
Aluminum Alloys
Aluminum Alloys
Aluminum Alloys
Aluminum Alloys
Beryllium Copper
Beryllium Copper
Beryllium Copper
Beryllium Copper
Beryllium Copper
Beryllium Copper
Beryllium Copper
Beryllium Copper
Beryllium Copper
Boron/ Epoxy
Boron/ Epoxy
Boron/ Epoxy
Boron/ Epoxy
Boron/ Epoxy
Boron/ Epoxy
Boron/ Epoxy
Boron/ Epoxy
Boron/ Epoxy
Cast Iron Alloys
Cast Iron Alloys
Cast Iron Alloys
Cast Iron Alloys
Cast Iron Alloys
Cast Iron Alloys
Cast Iron Alloys
Cast Iron Alloys
Cast Iron Alloys
Glass (Fused Silica)
Glass (Fused Silica)
Glass (Fused Silica)
Glass (Fused Silica)
Glass (Fused Silica)
Glass (Fused Silica)
Glass (Fused Silica)
Glass (Fused Silica)
Glass (Fused Silica)
Graphite/ Epoxy
Graphite/ Epoxy
Graphite/ Epoxy
Graphite/ Epoxy
Graphite/ Epoxy
Graphite/ Epoxy
Graphite/ Epoxy
Graphite/ Epoxy
Graphite/ Epoxy
Nickel Steel
Nickel Steel
Nickel Steel
Nickel Steel
Nickel Steel
Nickel Steel
Nickel Steel
Nickel Steel
Nickel Steel
Beryllium Copper
Boron/ Epoxy
Cast Iron Alloys
Glass (Fused Silica)
Graphite/ Epoxy
Nickel Steel
Polymethyl Methacrylate (PMMA)
Stainless Steel
Titanium
Aluminum Alloys
Boron/ Epoxy
Cast Iron Alloys
Glass (Fused Silica)
Graphite/ Epoxy
Nickel Steel
Polymethyl Methacrylate (PMMA)
Stainless Steel
Titanium
Aluminum Alloys
Beryllium Copper
Cast Iron Alloys
Glass (Fused Silica)
Graphite/ Epoxy
Nickel Steel
Polymethyl Methacrylate (PMMA)
Stainless Steel
Titanium
Aluminum Alloys
Beryllium Copper
Boron/ Epoxy
Glass (Fused Silica)
Graphite/ Epoxy
Nickel Steel
Polymethyl Methacrylate (PMMA)
Stainless Steel
Titanium
Aluminum Alloys
Beryllium Copper
Boron/ Epoxy
Cast Iron Alloys
Graphite/ Epoxy
Nickel Steel
Polymethyl Methacrylate (PMMA)
Stainless Steel
Titanium
Aluminum Alloys
Beryllium Copper
Boron/ Epoxy
Cast Iron Alloys
Glass (Fused Silica)
Nickel Steel
Polymethyl Methacrylate (PMMA)
Stainless Steel
Titanium
Aluminum Alloys
Beryllium Copper
Boron/ Epoxy
Cast Iron Alloys
Glass (Fused Silica)
Graphite/ Epoxy
Polymethyl Methacrylate (PMMA)
Stainless Steel
Titanium
5.04
5.04
5.04
5.04
5.04
5.04
5.04
5.04
5.04
3.90
3.90
3.90
3.90
3.90
3.90
3.90
3.90
3.90
9.87
9.87
9.87
9.87
9.87
9.87
9.87
9.87
9.87
5.73
5.73
5.73
5.73
5.73
5.73
5.73
5.73
5.73
5.72
5.72
5.72
5.72
5.72
5.72
5.72
5.72
5.72
9.10
9.10
9.10
9.10
9.10
9.10
9.10
9.10
9.10
4.91
4.91
4.91
4.91
4.91
4.91
4.91
4.91
4.91
3.90
9.87
5.73
5.72
9.10
4.91
2.52
4.96
5.03
5.04
9.87
5.73
5.72
9.10
4.91
2.52
4.96
5.03
5.04
3.90
5.73
5.72
9.10
4.91
2.52
4.96
5.03
5.04
3.90
9.87
5.72
9.10
4.91
2.52
4.96
5.03
5.04
3.90
9.87
5.73
9.10
4.91
2.52
4.96
5.03
5.04
3.90
9.87
5.73
5.72
4.91
2.52
4.96
5.03
5.04
3.90
9.87
5.73
5.72
9.10
2.52
4.96
5.03
Grouped by Ratio of Segment
Cross-Sectional Areas
70.3
70.3
70.3
70.3
70.3
70.3
70.3
70.3
70.3
124.1
124.1
124.1
124.1
124.1
124.1
124.1
124.1
124.1
195.0
195.0
195.0
195.0
195.0
195.0
195.0
195.0
195.0
240.0
240.0
240.0
240.0
240.0
240.0
240.0
240.0
240.0
72.0
72.0
72.0
72.0
72.0
72.0
72.0
72.0
72.0
206.8
206.8
206.8
206.8
206.8
206.8
206.8
206.8
206.8
199.9
199.9
199.9
199.9
199.9
199.9
199.9
199.9
199.9
124.1
195.0
240.0
72.0
206.8
199.9
7.6
190.3
113.8
70.3
195.0
240.0
72.0
206.8
199.9
7.6
190.3
113.8
70.3
124.1
240.0
72.0
206.8
199.9
7.6
190.3
113.8
70.3
124.1
195.0
72.0
206.8
199.9
7.6
190.3
113.8
70.3
124.1
195.0
240.0
206.8
199.9
7.6
190.3
113.8
70.3
124.1
195.0
240.0
72.0
199.9
7.6
190.3
113.8
70.3
124.1
195.0
240.0
72.0
206.8
7.6
190.3
113.8
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
-0.391
-0.172
-0.500
0.051
-0.240
-0.489
0.644
-0.467
-0.237
0.390
0.234
-0.136
0.433
0.166
-0.122
0.827
-0.093
0.169
0.172
-0.234
-0.359
0.221
-0.070
-0.346
0.735
-0.320
-0.068
0.500
0.136
0.359
0.538
0.296
0.015
0.865
0.044
0.298
-0.052
-0.433
-0.221
-0.538
-0.287
-0.527
0.613
-0.506
-0.285
0.239
-0.167
0.070
-0.296
0.287
-0.283
0.765
-0.256
0.002
0.489
0.122
0.347
-0.014
0.527
0.283
0.862
0.030
0.285
0.609
0.828
0.500
1.051
0.760
0.511
1.644
0.533
0.763
1.390
1.234
0.864
1.433
1.166
0.878
1.827
0.907
1.169
1.172
0.766
0.641
1.221
0.930
0.654
1.735
0.680
0.932
1.500
1.136
1.359
1.538
1.296
1.015
1.865
1.044
1.298
0.948
0.567
0.779
0.462
0.713
0.473
1.613
0.494
0.715
1.239
0.833
1.070
0.704
1.287
0.717
1.765
0.744
1.002
1.489
1.122
1.347
0.986
1.527
1.283
1.862
1.030
1.285
95
Tabulated Reflection and Transmission Coefficients
Polymethyl Methacrylate
Polymethyl Methacrylate
Polymethyl Methacrylate
Polymethyl Methacrylate
Polymethyl Methacrylate
Polymethyl Methacrylate
Polymethyl Methacrylate
Polymethyl Methacrylate
Polymethyl Methacrylate
Stainless Steel
Stainless Steel
Stainless Steel
Stainless Steel
Stainless Steel
Stainless Steel
Stainless Steel
Stainless Steel
Stainless Steel
Titanium
Titanium
Titanium
Titanium
Titanium
Titanium
Titanium
Titanium
Titanium
(PMMA)
(PMMA)
(PMMA)
(PMMA)
(PMMA)
(PMMA)
(PMMA)
(PMMA)
(PMMA)
Aluminum Alloys
Beryllium Copper
Boron/ Epoxy
Cast Iron Alloys
Glass (Fused Silica)
Graphite/ Epoxy
Nickel Steel
Stainless Steel
Titanium
Aluminum Alloys
Beryllium Copper
Boron/ Epoxy
Cast Iron Alloys
Glass (Fused Silica)
Graphite/ Epoxy
Nickel Steel
Polymethyl Methacrylate (PMMA)
Titanium
Aluminum Alloys
Beryllium Copper
Boron/ Epoxy
Cast Iron Alloys
Glass (Fused Silica)
Graphite/ Epoxy
Nickel Steel
Polymethyl Methacrylate (PMMA)
Stainless Steel
2.52
2.52
2.52
2.52
2.52
2.52
2.52
2.52
2.52
4.96
4.96
4.96
4.96
4.96
4.96
4.96
4.96
4.96
5.03
5.03
5.03
5.03
5.03
5.03
5.03
5.03
5.03
5.04
3.90
9.87
5.73
5.72
9.10
4.91
4.96
5.03
5.04
3.90
9.87
5.73
5.72
9.10
4.91
2.52
5.03
5.04
3.90
9.87
5.73
5.72
9.10
4.91
2.52
4.96
Grouped by Ratio of Segment
Cross-Sectional Areas
7.6
7.6
7.6
7.6
7.6
7.6
7.6
7.6
7.6
190.3
190.3
190.3
190.3
190.3
190.3
190.3
190.3
190.3
113.8
113.8
113.8
113.8
113.8
113.8
113.8
113.8
113.8
70.3
124.1
195.0
240.0
72.0
206.8
199.9
190.3
113.8
70.3
124.1
195.0
240.0
72.0
206.8
199.9
7.6
113.8
70.3
124.1
195.0
240.0
72.0
206.8
199.9
7.6
190.3
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
-0.645
-0.827
-0.735
-0.866
-0.614
-0.766
-0.862
-0.854
-0.765
0.467
0.093
0.320
-0.044
0.506
0.256
-0.029
0.854
0.258
0.237
-0.169
0.068
-0.298
0.285
-0.002
-0.285
0.764
-0.258
0.355
0.173
0.265
0.134
0.386
0.234
0.138
0.146
0.235
1.467
1.093
1.320
0.956
1.506
1.256
0.971
1.854
1.258
1.237
0.831
1.068
0.702
1.285
0.998
0.715
1.764
0.742
96
Tabulated Reflection and Transmission Coefficients
Aluminum Alloys
Aluminum Alloys
Aluminum Alloys
Aluminum Alloys
Aluminum Alloys
Aluminum Alloys
Aluminum Alloys
Aluminum Alloys
Aluminum Alloys
Beryllium Copper
Beryllium Copper
Beryllium Copper
Beryllium Copper
Beryllium Copper
Beryllium Copper
Beryllium Copper
Beryllium Copper
Beryllium Copper
Boron/ Epoxy
Boron/ Epoxy
Boron/ Epoxy
Boron/ Epoxy
Boron/ Epoxy
Boron/ Epoxy
Boron/ Epoxy
Boron/ Epoxy
Boron/ Epoxy
Cast Iron Alloys
Cast Iron Alloys
Cast Iron Alloys
Cast Iron Alloys
Cast Iron Alloys
Cast Iron Alloys
Cast Iron Alloys
Cast Iron Alloys
Cast Iron Alloys
Glass (Fused Silica)
Glass (Fused Silica)
Glass (Fused Silica)
Glass (Fused Silica)
Glass (Fused Silica)
Glass (Fused Silica)
Glass (Fused Silica)
Glass (Fused Silica)
Glass (Fused Silica)
Graphite/ Epoxy
Graphite/ Epoxy
Graphite/ Epoxy
Graphite/ Epoxy
Graphite/ Epoxy
Graphite/ Epoxy
Graphite/ Epoxy
Graphite/ Epoxy
Graphite/ Epoxy
Nickel Steel
Nickel Steel
Nickel Steel
Nickel Steel
Nickel Steel
Nickel Steel
Nickel Steel
Nickel Steel
Nickel Steel
Beryllium Copper
Boron/ Epoxy
Cast Iron Alloys
Glass (Fused Silica)
Graphite/ Epoxy
Nickel Steel
Polymethyl Methacrylate (PMMA)
Stainless Steel
Titanium
Aluminum Alloys
Boron/ Epoxy
Cast Iron Alloys
Glass (Fused Silica)
Graphite/ Epoxy
Nickel Steel
Polymethyl Methacrylate (PMMA)
Stainless Steel
Titanium
Aluminum Alloys
Beryllium Copper
Cast Iron Alloys
Glass (Fused Silica)
Graphite/ Epoxy
Nickel Steel
Polymethyl Methacrylate (PMMA)
Stainless Steel
Titanium
Aluminum Alloys
Beryllium Copper
Boron/ Epoxy
Glass (Fused Silica)
Graphite/ Epoxy
Nickel Steel
Polymethyl Methacrylate (PMMA)
Stainless Steel
Titanium
Aluminum Alloys
Beryllium Copper
Boron/ Epoxy
Cast Iron Alloys
Graphite/ Epoxy
Nickel Steel
Polymethyl Methacrylate (PMMA)
Stainless Steel
Titanium
Aluminum Alloys
Beryllium Copper
Boron/ Epoxy
Cast Iron Alloys
Glass (Fused Silica)
Nickel Steel
Polymethyl Methacrylate (PMMA)
Stainless Steel
Titanium
Aluminum Alloys
Beryllium Copper
Boron/ Epoxy
Cast Iron Alloys
Glass (Fused Silica)
Graphite/ Epoxy
Polymethyl Methacrylate (PMMA)
Stainless Steel
Titanium
5.04
5.04
5.04
5.04
5.04
5.04
5.04
5.04
5.04
3.90
3.90
3.90
3.90
3.90
3.90
3.90
3.90
3.90
9.87
9.87
9.87
9.87
9.87
9.87
9.87
9.87
9.87
5.73
5.73
5.73
5.73
5.73
5.73
5.73
5.73
5.73
5.72
5.72
5.72
5.72
5.72
5.72
5.72
5.72
5.72
9.10
9.10
9.10
9.10
9.10
9.10
9.10
9.10
9.10
4.91
4.91
4.91
4.91
4.91
4.91
4.91
4.91
4.91
3.90
9.87
5.73
5.72
9.10
4.91
2.52
4.96
5.03
5.04
9.87
5.73
5.72
9.10
4.91
2.52
4.96
5.03
5.04
3.90
5.73
5.72
9.10
4.91
2.52
4.96
5.03
5.04
3.90
9.87
5.72
9.10
4.91
2.52
4.96
5.03
5.04
3.90
9.87
5.73
9.10
4.91
2.52
4.96
5.03
5.04
3.90
9.87
5.73
5.72
4.91
2.52
4.96
5.03
5.04
3.90
9.87
5.73
5.72
9.10
2.52
4.96
5.03
Grouped by Ratio of Segment
Cross-Sectional Areas
70.3
70.3
70.3
70.3
70.3
70.3
70.3
70.3
70.31
124.1
124.1
124.1
124.1
124.1
124.1
124.1
124.1
124.1
195.0
195.0
195.0
195.0
195.0
195.0
195.0
195.0
195.0
240.0
240.0
240.0
240.0
240.0
240.0
240.0
240.0
240.0
72.0
72.0
72.0
72.0
72.0
72.0
72.0
72.0
72.0
206.8
206.8
206.8
206.8
206.8
206.8
206.8
206.8
206.8
199.9
199.9
199.9
199.9
199.9
199.9
199.9
199.9
199.9
124.1
195.0
240.0
72.0
206.8
199.9
7.6
190.3
113.8
70.3
195.0
240.0
72.0
206.8
199.9
7.6
190.3
113.8
70.3
124.1
240.0
72.0
206.8
199.9
7.6
190.3
113.8
70.3
124.1
195.0
72.0
206.8
199.9
7.6
190.3
113.8
70.3
124.1
195.0
240.0
206.8
199.9
7.6
190.3
113.8
70.3
124.1
195.0
240.0
72.0
199.9
7.6
190.3
113.8
70.3
124.1
195.0
240.0
72.0
206.8
7.6
190.3
113.8
10.0
10.0
10.0
10.0
10.0
10.0
10.0
10.0
10.0
10.0
10.0
10.0
10.0
10.0
10.0
10.0
10.0
10.0
10.0
10.0
10.0
10.0
10.0
10.0
10.0
10.0
10.0
10.0
10.0
10.0
10.0
10.0
10.0
10.0
10.0
10.0
10.0
10.0
10.0
10.0
10.0
10.0
10.0
10.0
10.0
10.0
10.0
10.0
10.0
10.0
10.0
10.0
10.0
10.0
10.0
10.0
10.0
10.0
10.0
10.0
10.0
10.0
10.0
0.628
0.752
0.538
0.834
0.720
0.548
0.958
0.568
0.721
0.916
0.883
0.768
0.924
0.867
0.773
0.981
0.785
0.867
0.868
0.722
0.650
0.880
0.794
0.659
0.970
0.675
0.794
0.936
0.859
0.910
0.942
0.897
0.823
0.986
0.832
0.897
0.800
0.596
0.729
0.501
0.694
0.512
0.953
0.533
0.695
0.884
0.754
0.840
0.689
0.895
0.696
0.974
0.711
0.819
0.934
0.855
0.907
0.814
0.940
0.894
0.985
0.828
0.895
1.628
1.752
1.538
1.834
1.720
1.548
1.958
1.568
1.721
1.916
1.883
1.768
1.924
1.867
1.773
1.981
1.785
1.867
1.868
1.722
1.650
1.880
1.794
1.659
1.970
1.675
1.794
1.936
1.859
1.910
1.942
1.897
1.823
1.986
1.832
1.897
1.800
1.596
1.729
1.501
1.694
1.512
1.953
1.533
1.695
1.884
1.754
1.840
1.689
1.895
1.696
1.974
1.711
1.819
1.934
1.855
1.907
1.814
1.940
1.894
1.985
1.828
1.895
97
Tabulated Reflection and Transmission Coefficients
Polymethyl Methacrylate (PMMA)
Polymethyl Methacrylate (PMMA)
Polymethyl Methacrylate (PMMA)
Polymethyl Methacrylate (PMMA)
Polymethyl Methacrylate (PMMA)
Polymethyl Methacrylate (PMMA)
Polymethyl Methacrylate (PMMA)
Polymethyl Methacrylate (PMMA)
Polymethyl Methacrylate (PMMA)
Stainless Steel
Stainless Steel
Stainless Steel
Stainless Steel
Stainless Steel
Stainless Steel
Stainless Steel
Stainless Steel
Stainless Steel
Titanium
Titanium
Titanium
Titanium
Titanium
Titanium
Titanium
Titanium
Titanium
I __ __
Aluminum Alloys
Beryllium Copper
Boron/ Epoxy
Cast Iron Alloys
Glass (Fused Silica)
Graphite/ Epoxy
Nickel Steel
Stainless Steel
Titanium
Aluminum Alloys
Beryllium Copper
Boron/ Epoxy
Cast Iron Alloys
Glass (Fused Silica)
Graphite/ Epoxy
Nickel Steel
Polymethyl Methacrylate (PMMA)
Titanium
Aluminum Alloys
Beryllium Copper
Boron/ Epoxy
Cast Iron Alloys
Glass (Fused Silica)
Graphite/ Epoxy
Nickel Steel
Polymethyl Methacrylate (PMMA)
Stainless Steel
2.52
2.52
2.52
2.52
2.52
2.52
2.52
2.52
2.52
4.96
4.96
4.96
4.96
4.96
4.96
4.96
4.96
4.96
5.03
5.03
5.03
5.03
5.03
5.03
5.03
5.03
5.03
5.04
3.90
9.87
5.73
5.72
9.10
4.91
4.96
5.03
5.04
3.90
9.87
5.73
5.72
9.10
4.91
2.52
5.03
5.04
3.90
9.87
5.73
5.72
9.10
4.91
2.52
4.96
Grouped by Ratio of Segment
Cross-Sectional Areas
7.6
7.6
7.6
7.6
7.6
7.6
7.6
7.6
7.6
190.3
190.3
190.3
190.3
190.3
190.3
190.3
190.3
190.3
113.8
113.8
113.8
113.8
113.8
113.8
113.8
113.8
113.8
70.3
124.1
195.0
240.0
72.0
206.8
199.9
190.3
113.8
70.3
124.1
195.0
240.0
72.0
206.8
199.9
7.6
113.8
70.3
124.1
195.0
240.0
72.0
206.8
199.9
7.6
190.3
10.0
10.0
10.0
10.0
10.0
10.0
10.0
10.0
10.0
10.0
10.0
10.0
10.0
10.0
10.0
10.0
10.0
10.0
10.0
10.0
10.0
10.0
10.0
10.0
10.0
10.0
10.0
0.367
-0.027
0.209
-0.162
0.411
0.140
-0.148
-0.120
0.143
0.930
0.847
0.902
0.803
0.936
0.888
0.808
0.984
0.889
0.884
0.753
0.839
0.688
0.895
0.817
0.695
0.974
0.710
1.367
0.973
1.209
0.838
1.411
1.140
0.852
0.880
1.143
1.930
1.847
1.902
1.803
1.936
1.888
1.808
1.984
1.889
1.884
1.753
1.839
1.688
1.895
1.817
1.695
1.974
1.710
98
Tabulated Reflection and Transmission Coefficients
Aluminum Alloys
Aluminum Alloys
Aluminum Alloys
Aluminum Alloys
Aluminum Alloys
Aluminum Alloys
Aluminum Alloys
Aluminum Alloys
Aluminum Alloys
Beryllium Copper
Beryllium Copper
Beryllium Copper
Beryllium Copper
Beryllium Copper
Beryllium Copper
Beryllium.Copper
Beryllium Copper
Beryllium Copper
Boron/ Epoxy
Boron/ Epoxy
Boron/ Epoxy
Boron/ Epoxy
Boron/ Epoxy
Boron/ Epoxy
Boron/ Epoxy
Boron/ Epoxy
Boron/ Epoxy
Cast Iron Alloys
Cast Iron Alloys
Cast Iron Alloys
Cast Iron Alloys
Cast Iron Alloys
Cast Iron Alloys
Cast Iron Alloys
Cast Iron Alloys
Cast Iron Alloys
Glass (Fused Silica)
Glass (Fused Silica)
Glass (Fused Silica)
Glass (Fused Silica)
Glass (Fused Silica)
Glass (Fused Silica)
Glass (Fused Silica)
Glass (Fused Silica)
Glass (Fused Silica)
Graphite/ Epoxy
Graphite/ Epoxy
Graphite/ Epoxy
Graphite/ Epoxy
Graphite/ Epoxy
Graphite/ Epoxy
Graphite/ Epoxy
Graphite/ Epoxy
Graphite/ Epoxy
Nickel Steel
Nickel Steel
Nickel Steel
Nickel Steel
Nickel Steel
Nickel Steel
Nickel Steel
Nickel Steel
Nickel Steel
Beryllium Copper
Boron/ Epoxy
Cast Iron Alloys
Glass (Fused Silica)
Graphite/ Epoxy
Nickel Steel
Polymethyl Methacrylate (PMMA)
Stainless Steel
Titanium
Aluminum Alloys
Boron/ Epoxy
Cast Iron Alloys
Glass (Fused Silica)
Graphite/ Epoxy
Nickel Steel
Polymethyl Methacrylate (PMMA)
Stainless Steel
Titanium
Aluminum Alloys
Beryllium Copper
Cast Iron Alloys
Glass (Fused Silica)
Graphite/ Epoxy
Nickel Steel
Polymethyl Methacrylate (PMMA)
Stainless Steel
Titanium
Aluminum Alloys
Beryllium Copper
Boron/ Epoxy
Glass (Fused Silica)
Graphite/ Epoxy
Nickel Steel
Polymethyl Methacrylate (PMMA)
Stainless Steel
Titanium
Aluminum Alloys
Beryllium Copper
Boron/ Epoxy
Cast Iron Alloys
Graphite/ Epoxy
Nickel Steel
Polymethyl Methacrylate (PMMA)
Stainless Steel
Titanium
Aluminum Alloys
Beryllium Copper
Boron/ Epoxy
Cast Iron Alloys
Glass (Fused Silica)
Nickel Steel
Polymethyl Methacrylate (PMMA)
Stainless Steel
Titanium
Aluminum Alloys
Beryllium Copper
Boron/ Epoxy
Cast Iron Alloys
Glass (Fused Silica)
Graphite/ Epoxy
Polymethyl Methacrylate (PMMA)
Stainless Steel
Titanium
5.04
5.04
5.04
5.04
5.04
5.04
5.04
5.04
5.04
3.90
3.90
3.90
3.90
3.90
3.90
3.90
3.90
3.90
9.87
9.87
9.87
9.87
9.87
9.87
9.87
9.87
9.87
5.73
5.73
5.73
5.73
5.73
5.73
5.73
5.73
5.73
5.72
5.72
5.72
5.72
5.72
5.72
5.72
5.72
5.72
9.10
9.10
9.10
9.10
9.10
9.10
9.10
9.10
9.10
4.91
4.91
4.91
4.91
4.91
4.91
4.91
4.91
4.91
3.90
9.87
5.73
5.72
9.10
4.91
2.52
4.96
5.03
5.04
9.87
5.73
5.72
9.10
4.91
2.52
4.96
5.03
5.04
3.90
5.73
5.72
9.10
4.91
2.52
4.96
5.03
5.04
3.90
9.87
5.72
9.10
4.91
2.52
4.96
5.03
5.04
3.90
9.87
5.73
9.10
4.91
2.52
4.96
5.03
5.04
3.90
9.87
5.73
5.72
4.91
2.52
4.96
5.03
5.04
3.90
9.87
5.73
5.72
9.10
2.52
4.96
5.03
Grouped by Ratio of Segment
Cross-Sectional Areas
70.3
70.3
70.3
70.3
70.3
70.3
70.3
70.3
70.3
124.1
124.1
124.1
124.1
124.1
124.1
124.1
124.1
124.1
195.0
195.0
195.0
195.0
195.0
195.0
195.0
195.0
195.0
240.0
240.0
240.0
240.0
240.0
240.0
240.0
240.0
240.0
72.0
72.0
72.0
72.0
72.0
72.0
72.0
72.0
72.0
206.8
206.8
206.8
206.8
206.8
206.8
206.8
206.8
206.8
199.9
199.9
199.9
199.9
199.9
199.9
199.9
199.9
199.9
124.1
195.0
240.0
72.0
206.8
199.9
7.6
190.3
113.8
70.3
195.0
240.0
72.0
206.8
199.9
7.6
190.3
113.8
70.3
124.1
240.0
72.0
206.8
199.9
7.6
190.3
113.8
70.3
124.1
195.0
72.0
206.8
199.9
7.6
190.3
113.8
70.3
124.1
195.0
240.0
206.8
199.9
7.6
190.3
113.8
70.3
124.1
195.0
240.0
72.0
199.9
7.6
190.3
113.8
70.3
124.1
195.0
240.0
72.0
206.8
7.6
190.3
113.8
100.0
100.0
100.0
100.0
100.0
100.0
100.0
100,0
100.0
100.0
100.0
100.0
100.0
100.0
100.0
100.0
100.0
100.0
100.0
100.0
100.0
100.0
100.0
100.0
100.0
100.0
100.0
100.0
100.0
100.0
100.0
100.0
100.0
100.0
100.0
100.0
100.0
100.0
100.0
100.0
100.0
100.0
100.0
100.0
100.0
100.0
100.0
100.0
100.0
100.0
100.0
100.0
100.0
100.0
100.0
100.0
100.0
100.0
100.0
100.0
100.0
100.0
100.0
0.955
0.972
0.942
0.982
0.968
0.943
0.996
0.946
0.968
0.991
0.988
0.974
0.992
0.986
0.975
0.998
0.976
0.986
0.986
0.968
0.959
0.987
0.977
0.960
0.997
0.962
0.977
0.993
0.985
0.991
0.994
0.989
0.981
0.999
0.982
0.989
0.978
0.951
0.969
0.936
0.965
0.937
0.995
0.941
0.965
0.988
0.972
0.983
0.964
0.989
0.965
0.997
0.967
0.980
0.993
0.984
0.990
0.980
0.994
0.989
0.999
0.981
0.989
1.955
1.972
1.942
1.982
1.968
1.943
1.996
1.946
1.968
1.991
1.988
1.974
1.992
1.986
1.975
1.998
1.976
1.986
1.986
1.968
1.959
1.987
1.977
1.960
1.997
1.962
1.977
1.993
1.985
1.991
1.994
1.989
1.981
1.999
1.982
1.989
1.978
1.951
1.969
1.936
1.965
1.937
1.995
1.941
1.965
1.988
1.972
1.983
1.964
1.989
1.965
1.997
1.967
1.980
1.993
1.984
1.990
1.980
1.994
1.989
1.999
1.981
1.989
99
Tabulated Reflection and Transmission Coefficients
Polymethyl Methacrylate (PMMA)
Polymethyl Methacrylate (PMMA)
Polymethyl Methacrylate (PMMA)
Polymethyl Methacrylate (PMMA)
Polymethyl Methacrylate (PMMA)
Polymethyl Methacrylate (PMMA)
Polymethyl Methacrylate (PMMA)
Polymethyl Methacrylate (PMMA)
Polymethyl Methacrylate (PMMA)
Stainless Steel
Stainless Steel
Stainless Steel
Stainless Steel
Stainless Steel
Stainless Steel
Stainless Steel
Stainless Steel
Stainless Steel
Titanium
Titanium
Titanium
Titanium
Titanium
Titanium
Titanium
Titanium
Titanium
Aluminum Alloys
Beryllium Copper
Boron/ Epoxy
Cast Iron Alloys
Glass (Fused Silica)
Graphite/ Epoxy
Nickel Steel
Stainless Steel
Titanium
Aluminum Alloys
Beryllium Copper
Boron/ Epoxy
Cast Iron Alloys
Glass (Fused Silica)
Graphite/ Epoxy
Nickel Steel
Polymethyl Metha crylate (PMMA)
Titanium
Aluminum Alloys
Beryllium Copper
Boron/ Epoxy
Cast Iron Alloys
Glass (Fused Silica]
Graphite/ Epoxy
Nickel Steel
Polymethyl Metha crylate (PMMA)
Stainless Steel
2.52
2.52
2.52
2.52
2.52
2.52
2.52
2.52
2.52
4.96
4.96
4.96
4.96
4.96
4.96
4.96
4.96
4.96
5.03
5.03
5.03
5.03
5.03
5.03
5.03
5.03
5.03
5.04
3.90
9.87
5.73
5.72
9.10
4.91
4.96
5.03
5.04
3.90
9.87
5.73
5.72
9.10
4.91
2.52
5.03
5.04
3.90
9.87
5.73
5.72
9.10
4.91
2.52
4.96
Grouped by Ratio of Segment
Cross-Sectional Areas
7.6
7.6
7.6
7.6
7.6
7.6
7.6
7.6
7.6
7.63
190.3
190.3
190.3
190.3
190.3
190.3
190.3
190.3
190.3
113.8
113.8
113.8
113.8
113.8
113.8
113.8
113.8
113.8
70.3
70.3
124.1
195.0
240.0
72.0
206.8
199.9
190.3
113.8
70.3
124.1
195.0
240.0
72.0
206.8
199.9
7.6
113.8
70.3
124.1
195.0
240.0
72.0
206.8
199.9
7.6
190.3
1uu.u
100.0
100.0
100.0
100.0
100.0
100.0
100.0
100.0
100.0
100.0
100.0
100.0
100.0
100.0
100.0
100.0
100.0
100.0
100.0
100.0
100.0
100.0
100.0
100.0
100.0
100.0
100.0
U.~I.LL
U.912
0.809
0.877
0.756
0.920
0.860
0.762
0.774
0.860
0.993
0.984
0.990
0.978
0.993
0.988
0.979
0.998
0.988
0.988
0.972
0.983
0.964
0.989
0.980
0.965
0.997
0.967
1.809
1.877
1.756
1.920
1.860
1.762
1.774
1.860
1.993
1.984
1.990
1.978
1.993
1988
1.979
1.998
1.988
1.988
1.972
1.983
1.964
1.989
1.980
1.965
1.997
1.967
100
Tabulated Reflection and Transmission Coefficients
7EL
Aluminum Alloys
Aluminum Alloys
Aluminum Alloys
Aluminum Alloys
Aluminum Alloys
Aluminum Alloys
Aluminum Alloys
Aluminum Alloys
Aluminum Alloys
Aluminum Alloys
Aluminum Alloys
Aluminum Alloys
Aluminum Alloys
Aluminum Alloys
Aluminum Alloys
Aluminum Alloys
Aluminum Alloys
Aluminum Alloys
Aluminum Alloys
Aluminum Alloys
Aluminum Alloys
Aluminum Alloys
Aluminum Alloys
Aluminum Alloys
Aluminum Alloys
Aluminum Alloys
Aluminum Alloys
Aluminum Alloys
Aluminum Alloys
Aluminum Alloys
Aluminum Alloys
Aluminum Alloys
Aluminum Alloys
Aluminum Alloys
Aluminum Alloys
Aluminum Alloys
Beryllium Copper
Beryllium Copper
Beryllium Copper
Beryllium Copper
Beryllium Copper
Beryllium Copper
Beryllium Copper
Beryllium Copper
Beryllium Copper
Beryllium Copper
Beryllium Copper
Beryllium Copper
Beryllium Copper
Beryllium Copper
Beryllium Copper
Beryllium Copper
Beryllium Copper
Beryllium Copper
Beryllium Copper
Beryllium Copper
Beryllium Copper
Beryllium Copper
Beryllium Copper
Beryllium Copper
Beryllium Copper
Beryllium Copper
Beryllium Copper
J,
Beryllium Copper
Beryllium Copper
Beryllium Copper
Beryllium Copper
Boron/ Epoxy
Boron/ Epoxy
Boron/ Epoxy
Boron/ Epoxy
Cast Iron Alloys
Cast Iron Alloys
Cast Iron Alloys
Cast Iron Alloys
Glass (Fused Silica)
Glass (Fused Silica)
Glass (Fused Silica)
Glass (Fused Silica)
Graphite/ Epoxy
Graphite/ Epoxy
Graphite/ Epoxy
Graphite/ Epoxy
Nickel Steel
Nickel Steel
Nickel Steel
Nickel Steel
Polymethyl Methacrylate (PMMA)
Polymethyl Methacrylate (PMMA)
Polymethyl Methacrylate (PMMA)
Polymethyl Methacrylate (PMMA)
Stainless Steel
Stainless Steel
Stainless Steel
Stainless Steel
Titanium
Titanium
Titanium
Titanium
Aluminum Alloys
Aluminum Alloys
Aluminum Alloys
Aluminum Alloys
Boron/ Epoxy
Boron/ Epoxy
Boron/ Epoxy
Boron/ Epoxy
Cast Iron Alloys
Cast Iron Alloys
Cast Iron Alloys
Cast Iron Alloys
Glass (Fused Silica)
Glass (Fused Silica)
Glass (Fused Silica)
Glass (Fused Silica)
Graphite/ Epoxy
Graphite/ Epoxy
Graphite/ Epoxy
Graphite/ Epoxy
Nickel Steel
Nickel Steel
Nickel Steel
Nickel Steel
Polymethyl Methacrylate (PMMA)
Polymethyl Methacrylate (PMMA)
Polymethyl Methacrylate (PMMA)
5.04
5.04
5.04
5.04
5.04
5.04
5.04
5.04
5.04
5.04
5.04
5.04
5.04
5.04
5.04
5.04
5.04
5.04
5.04
5.04
5.04
5.04
5.04
5.04
5.04
5.04
5.04
5.04
5.04
5.04
5.04
5.04
5.04
5.04
5.04
5.04
3.90
3.90
3.90
3.90
3.90
3.90
3.90
3.90
3.90
3.90
3.90
3.90
3.90
3.90
3.90
3.90
3.90
3.90
3.90
3.90
3.90
3.90
3.90
3.90
3.90
3.90
3.90
3.90
3.90
3.90
3.90
9.87
9.87
9.87
9.87
5.73
5.73
5.73
5.73
5.72
5.72
5.72
5.72
9.10
9.10
9.10
9.10
4.91
4.91
4.91
4.91
2.52
2.52
2.52
2.52
4.96
4.96
4.96
4.96
5.03
5.03
5.03
5.03
5.04
5.04
5.04
5.04
9.87
9.87
9.87
9.87
5.73
5.73
5.73
5.73
5.72
5.72
5.72
5.72
9.10
9.10
9.10
9.10
4.91
4.91
4.91
4.91
2.52
2.52
2.52
Grouped by Material Pair
70.3
70.3
70.3
70.3
70.3
70.3
70.3
70.3
70.3
70.3
70.3
70.3
70.3
70.3
70.3
70.3
70.3
70.3
70.3
70.3
70.3
70.3
70.3
70.3
70.3
70.3
70.3
70.3
70.3
70.3
70.3
70.3
70.3
70.3
70.3
70.3
124.1
124.1
124.1
124.1
124.1
124.1
124.1
124.1
124.1
124.1
124.1
124.1
124.1
124.1
124.1
124.1
124.1
124.1
124.1
124.1
124.1
124.1
124.1
124.1
124.1
124.1
124.1
124.1
124.1
124.1
124.1
195.0
195.0
195.0
195.0
240.0
240.0
240.0
240.0
72.0
72.0
72.0
72.0
206.8
206.8
206.8
206.8
199.9
199.9
199.9
199.9
7.6
7.6
7.6
7.6
190.3
190.3
190.3
190.3
113.8
113.8
113.8
113.8
70.3
70.3
70.3
70.3
195.0
195.0
195.0
195.0
240.0
240.0
240.0
240.0
72.0
72.0
72.0
72.0
206.8
206.8
206.8
206.8
199.9
199.9
199.9
199.9
7.6
7.6
7.6
0.1
1.0
10.0
100.0
0.1
1.0
10.0
100.0
0.1
1.0
10.0
100.0
0.1
1.0
10.0
100.0
0.1
1.0
10.0
100.0
0.1
1.0
10.0
100.0
0.1
1.0
10.0
100.0
0.1
1.0
10.0
100.0
0.1
1.0
10.0
100.0
0.1
1.0
10.0
100.0
0.1
1.0
10.0
100.0
0.1
1.0
10.0
100.0
0.1
1.0
10.0
100.0
0.1
1.0
10.0
100.0
0.1
1.0
10.0
100.0
0.1
1.0
10.0
-0.916
-0.34
0.628
0.955
-0.868
-0.172
0.752
0.972
-0.936
-0.500
0.538
0.942
-0.801
0.051
0.834
0.982
-0.884
-0.240
0.720
0.968
-0.934
-0.489
0.548
0.943
-0.368
0.644
0.958
0.996
-0.930
-0.467
0.568
0.946
-0.884
-0.237
0.721
0.968
-0.629
0.390
0.916
0.991
-0.722
0.234
0.883
0.988
-0.859
-0.136
0.768
0.974
-0.597
0.433
0.924
0.992
-0.754
0.166
0.867
0.986
-0.855
-0.122
0.773
0.975
0.026
0.827
0.981
0.084
0.609
1.628
1.955
0.132
0.828
1.752
1.972
0.064
0.500
1.538
1.942
0.199
1.051
1.834
1.982
0.116
0.760
1.720
1.968
0.066
0.511
1.548
1.943
0.632
1.644
1.958
1.996
0.070
0.533
1.568
1.946
0.116
0.763
1.721
1.968
0.371
1.390
1.916
1.991
0.278
1.234
1.883
1.988
0.141
0.864
1.768
1.974
0.403
1.433
1.924
1.992
0.246
1.166
1.867
1.986
0.145
0.878
1.773
1.975
1.026
1.827
1.981
101
Tabulated Reflection and Transmission Coefficients
Beryllium Copper
Beryllium Copper
Beryllium Copper
Beryllium Copper
Beryllium Copper
Beryllium Copper
Beryllium Copper
Beryllium Copper
Beryllium Copper
Boron/ Epoxy
Boron/ Epoxy
Boron/ Epoxy
Boron/ Epoxy
Boron/ Epoxy
Boron/ Epoxy
Boron/ Epoxy
Boron/ Epoxy
Boron/ Epoxy
Boron/ Epoxy
Boron/ Epoxy
Boron/ Epoxy
Boron/ Epoxy
Boron/ Epoxy
Boron/ Epoxy
Boron/ Epoxy
Boron/ Epoxy
Boron/ Epoxy
Boron/ Epoxy
Boron/ Epoxy
Boron/ Epoxy
Boron/ Epoxy
Boron/ Epoxy
Boron/ Epoxy
Boron/ Epoxy
Boron/ Epoxy
Boron/ Epoxy
Boron/ Epoxy
Boron/ Epoxy
Boron/ Epoxy
Boron/ Epoxy
Boron/ Epoxy
Boron/ Epoxy
Boron/ Epoxy
Boron/ Epoxy
Boron/ Epoxy
Cast Iron Alloys
Cast Iron Alloys
Cast Iron Alloys
Cast Iron Alloys
Cast Iron Alloys
Cast Iron Alloys
Cast Iron Alloys
Cast Iron Alloys
Cast Iron Alloys
Cast Iron Alloys
Cast Iron Alloys
Cast Iron Alloys
Cast Iron Alloys
Cast Iron Alloys
Cast Iron Alloys
Cast Iron Alloys
Cast Iron Alloys
Cast Iron Alloys
Polymethyl Methacrylate (PMMA)
Stainless Steel
Stainless Steel
Stainless Steel
Stainless Steel
Titanium
Titanium
Titanium
Titanium
Aluminum Alloys
Aluminum Alloys
Aluminum Alloys
Aluminum Alloys
Beryllium Copper
Beryllium Copper
Beryllium Copper
Beryllium Copper
Cast Iron Alloys
Cast Iron Alloys
Cast Iron Alloys
Cast Iron Alloys
Glass (Fused Silica)
Glass (Fused Silica)
Glass (Fused Silica)
Glass (Fused Silica)
Graphite/ Epoxy
Graphite/ Epoxy
Graphite/ Epoxy
Graphite/ Epoxy
Nickel Steel
Nickel Steel
Nickel Steel
Nickel Steel
Polymethyl Methacrylate (PMMA)
Polymethyl Methacrylate (PMMA)
Polymethyl Methacrylate (PMMA)
Polymethyl Methacrylate (PMMA)
Stainless Steel
Stainless Steel
Stainless Steel
Stainless Steel
Titanium
Titanium
Titanium
Titanium
Aluminum Alloys
Aluminum Alloys
Aluminum Alloys
Aluminum Alloys
Beryllium Copper
Beryllium Copper
Beryllium Copper
Beryllium Copper
Boron/ Epoxy
Boron/ Epoxy
Boron/ Epoxy
Boron/ Epoxy
Glass (Fused Silica)
Glass (Fused Silica)
Glass (Fused Silica)
Glass (Fused Silica)
Graphite/ Epoxy
Graphite/ Epoxy
3.90
3.90
3.90
3.90
3.90
3.90
3.90
3.90
3.90
9.87
9.87
9.87
9.87
9.87
9.87
9.87
9.87
9.87
9.87
9.87
9.87
9.87
9.87
9.87
9.87
9.87
9.87
9.87
9.87
9.87
9.87
9.87
9.87
9.87
9.87
9.87
9.87
9.87
9.87
9.87
9.87
9.87
9.87
9.87
9.87
5.73
5.73
5.73
5.73
5.73
5.73
5.73
5.73
5.73
5.73
5.73
5.73
5.73
5.73
5.73
5.73
5.73
5.73
2.52
4.96
4.96
4.96
4.96
5.03
5.03
5.03
5.03
5.04
5.04
5.04
5.04
3.90
3.90
3.90
3.90
5.73
5.73
5.73
5.73
5.72
5.72
5.72
5.72
9.10
9.10
9.10
9.10
4.91
4.91
4.91
4.91
2.52
2.52
2.52
2.52
4.96
4.96
4.96
4.96
5.03
5.03
5.03
5.03
5.04
5.04
5.04
5.04
3.90
3.90
3.90
3.90
9.87
9.87
9.87
9.87
5.72
5.72
5.72
5.72
9.10
9.10
Grouped by Material Pair
124.1
124.1
124.1
124.1
124.1
124.1
124.1
124.1
124.1
195.0
195.0
195.0
195.0
195.0
195.0
195.0
195.0
195.0
195.0
195.0
195.0
195.0
195.0
195.0
195.0
195.0
195.0
195.0
195.0
195.0
195.0
195.0
195.0
195.0
195.0
195.0
195.0
195.0
195.0
195.0
195.0
195.0
195.0
195.0
195.0
240.0
240.0
240.0
240.0
240.0
240.0
240.0
240.0
240.0
240.0.
240.0
240.0
240.0
240.0
240.0
240.0
240.0
240.0
7.6
190.3
190.3
190.3
190.3
113.8
113.8
113.8
113.8
70.3
70.3
70.3
70.3
124.1
124.1
124.1
124.1
240.0
240.0
240.0
240.0
72.0
72.0
72.0
72.0
206.8
206.8
206.8
206.8
199.9
199.9
199.9
199.9
7.6
7.6
7.6
7.6
190.3
190.3
190.3
190.3
113.8
113.8
113.8
113.8
70.3
70.3
70.3
70.3
124.1
124.1
124.1
124.1
195.0
19S.0
195.0
195.0
72.0
72.0
72.0
72.0
206.8
206.8
100.0
0.1
1.0
10.0
100.0
0.1
1.0
10.0
100.0
0.1
1.0
10.0
100.0
0.1
1.0
10.0
100.0
0.1
1.0
10.0
100.0
0.1
1.0
10.0
100.0
0.1
1.0
10.0
100.0
0.1
1.0
10.0
100.0
0.1
1.0
10.0
100.0
0.1
1.0
10.0
100.0
0.1
1.0
10.0
100.0
0.1
1.0
10.0
100.0
0.1
1.0
10.0
100.0
0.1
1.0
10.0
100.0
0.1
1.0
10.0
100.0
0.1
1.0
0.998
-0.847
-0.093
0.785
0.976
-0.753
0.169
0.867
0.986
-0.752
0.172
0.868
0.986
-0.883
-0.234
0.722
0.968
-0.910
-0.359
0.650
0.959
-0.729
0.221
0.880
0.987
-0.840
-0.070
0.794
0.977
-0.907
-0.346
0.659
0.960
-0.209
0.735
0.970
0.997
-0.902
-0.320
0.675
0.962
-0.839
-0.068
0.794
0.977
-0.538
0.500
0.936
0.993
-0.768
0.136
0.859
0.985
-0.650
0.359
0.910
0.991
-0.501
0.538
0.942
0.994
-0.689
0.296
1.998
0.153
0.907
1.785
1.976
0.247
1.169
1.867
1.986
0.248
1.172
1.868
1.986
0.117
0.766
1.722
1.968
0.090
0.641
1.650
1.959
0.271
1221
1.880
1.987
0.160
0.930
1.794
1.977
0.093
0.654
1.659
1.960
0.791
1.735
1.970
1.997
0.098
0.680
1.675
1.962
0.161
0.932
1.794
1.977
0.462
1.500
1.936
1.993
0.232
1.136
1.859
1.985
0.350
1.359
1.910
1.991
0.499
1.538
1.942
1.994
0.311
1.296
102
Tabulated Reflection and Transmission Coefficients
7. i1r.
7
Cast Iron Alloys
Cast Iron Alloys
Cast Iron Alloys
Cast Iron Alloys
Cast Iron Alloys
Cast Iron Alloys
Cast Iron Alloys
Cast Iron Alloys
Cast Iron Alloys
Cast Iron Alloys
Cast Iron Alloys
Cast Iron Alloys
Cast Iron Alloys
Cast Iron Alloys
Cast Iron Alloys
Cast Iron Alloys
Cast Iron Alloys
Cast Iron Alloys
Glass (Fused Silica)
Glass (Fused Silica)
Glass (Fused Silica)
Glass (Fused Silica)
Glass (Fused Silica)
Glass (Fused Silica)
Glass (Fused Silica)
Glass (Fused Silica)
Glass (Fused Silica)
Glass (Fused Silica)
Glass (Fused Silica)
Glass (Fused Silica)
Glass (Fused Silica)
Glass (Fused Silica)
Glass (Fused Silica)
Glass (Fused Silica)
Glass (Fused Silica)
Glass (Fused Silica)
Glass (Fused Silica)
Glass (Fused Silica)
Glass (Fused Silica)
Glass (Fused Silica)
Glass (Fused Silica)
Glass (Fused Silica)
Glass (Fused Silica)
Glass (Fused Silica)
Glass (Fused Silica)
Glass (Fused Silica)
Glass (Fused Silica)
Glass (Fused Silica)
Glass (Fused Silica)
Glass (Fused Silica)
Glass (Fused Silica)
Glass (Fused Silica)
Glass (Fused Silica)
Glass (Fused Silica)
Graphite/ Epoxy
Graphite/ Epoxy
Graphite/ Epoxy
Graphite/ Epoxy
Graphite/ Epoxy
Graphite/ Epoxy
Graphite/ Epoxy
Graphite/ Epoxy
Graphite/ Epoxy
Graphite/ Epoxy
Graphite/ Epoxy
Nickel Steel
Nickel Steel
Nickel Steel
Nickel Steel
Polymethyl Methacrylate (PMMA)
Polymethyl Methacrylate (PMMA)
Polymethyl Methacrylate (PMMA)
Polymethyl Methacrylate (PMMA)
Stainless Steel
Stainless Steel
Stainless Steel
Stainless Steel
Titanium
Titanium
Titanium
Titanium
Aluminum Alloys
Aluminum Alloys
Aluminum Alloys
Aluminum Alloys
Beryllium Copper
Beryllium Copper
Beryllium Copper
Beryllium Copper
Boron/ Epoxy
Boron/ Epoxy
Boron/ Epoxy
Boron/ Epoxy
Cast Iron Alloys
Cast Iron Alloys
Cast Iron Alloys
Cast Iron Alloys
Graphite/ Epoxy
Graphite/ Epoxy
Graphite/ Epoxy
Graphite/ Epoxy
Nickel Steel
Nickel Steel
Nickel Steel
Nickel Steel
Polymethyl Methacrylate (PMMA)
Polymethyl Methacrylate (PMMA)
Polymethyl Methacrylate (PMMA)
Polymethyl Methacrylate (PMMA)
Stainless Steel
Stainless Steel
Stainless Steel
Stainless Steel
Titanium
Titanium
Titanium
Titanium
Aluminum Alloys
Aluminum Alloys
Aluminum Alloys
Aluminum Alloys
Beryllium Copper
Beryllium Copper
Beryllium Copper
Beryllium Copper
Boron/ Epoxy
5.73
5.73
5.73
5.73
5.73
5.73
5.73
5.73
5.73
5.73
5.73
5.73
5.73
5.73
5.73
5.73
5.73
5.73
5.72
5.72
5.72
5.72
5.72
5.72
5.72
5.72
5.72
5.72
5.72
5.72
5.72
5.72
5.72
5.72
5.72
5.72
5.72
5.72
5.72
5.72
5.72
5.72
5.72
5.72
5.72
5.72
5.72
5.72
5.72
5.72
5.72
5.72
5.72
5.72
9.10
9.10
9.10
9.10
9.10
9.10
9.10
9.10
9.10
9.10
9.10
4.91
4.91
4.91
4.91
2.52
2.52
2.52
2.52
4.96
4.96
4.96
4.96
5.03
5.03
5.03
5.03
5.04
5.04
5.04
5.04
3.90
3.90
3.90
3.90
9.87
9.87
9.87
9.87
5.73
5.73
5.73
5.73
9.10
9.10
9.10
9.10
4.91
4.91
4.91
4.91
2.52
2.52
2.52
2.52
4.96
4.96
4.96
4.96
5.03
5.03
5.03
5.03
5.04
5.04
5.04
5.04
3.90
3.90
3.90
3.90
9.87
Grouped by Material Pair
240.0
240.0
240.0
240.0
240.0
240.0
240.0
240.0
240.0
240.0
240.0
240.0
240.0
240.0
240.0
240.0
240.0
240.0
72.0
72.0
72.0
72.0
72.0
72.0
72.0
72.0
72.0
72.0
72.0
72.0
72.0
72.0
72.0
72.0
72.0
72.0
72.0
72.0
72.0
72.0
72.0
72.0
72.0
72.0
72.0
72.0
72.0
72.0
72.0
72.0
72.0
72.0
72.0
72.0
206.8
206.8
206.8
206.8
206.8
206.8
206.8
206.8
206.8
206.8
206.8
199.9
199.9
199.9
199.9
7.6
7.6
7.6
7.6
190.3
190.3
190.3
190.3
113.8
113.8
113.8
113.8
70.3
70.3
70.3
70.3
124.1
124.1
124.1
124.1
195.0
195.0
195.0
195.0
240.0
240.0
240.0
240.0
206.8
206.8
206.8
206.8
199.9
199.9
199.9
199.9
7.6
7.6
7.6
7.6
190.3
190.3
190.3
190.3
113.8
113.8
113.8
113.8
70.3
70.3
70.3
70.3
124.1
124.1
124.1
124.1
195.0
10.0
100.0
0.1
1.0
10.0
100.0
0.1
1.0
10.0
100.0
0.1
1.0
10.0
100.0
0.1
1.0
10.0
100.0
0.1
1.0
10.0
100.0
0.1
1.0
10.0
100.0
0.1
1.0
10.0
100.0
0.1
1.0
10.0
100.0
0.1
1.0
10.0
100.0
0.1
1.0
10.0
100.0
0.1
1.0
10.0
100.0
0.1
1.0
10.0
100.0
0.1
1.0
10.0
100.0
0.1
1.0
10.0
100.0
0.1
1.0
10.0
100.0
0.1
0.897
0.989
-0.813
0.015
0.823
0.981
0.162
0.865
0.986
0.999
-0.803
0.044
0.832
0.982
-0.688
0.298
0.897
0.989
-0.835
-0.052
0.800
0.978
-0.924
-0.433
0.596
0.951
-0.880
-0.221
0.729
0.969
-0.942
-0.538
0.501
0.936
-0.895
-0.287
0.694
0.965
-0.940
-0.527
0.512
0.937
-0.412
0.613
0.953
0.995
-0.936
-0.506
0.533
0.941
-0.895
-0.285
0.695
0.965
-0.720
0.239
0.884
0.988
-0.867
-0.167
0.754
0.972
-0.794
1.897
1.989
0.187
1.015
1.823
1.981
1.162
1.865
1.986
1.999
0.197
1.044
1.832
1.982
0.312
1.298
1.897
1.989
0.165
0.948
1.800
1.978
0.076
0.567
1.596
1.951
0.120
0.779
1.729
1.969
0.058
0.462
1.501
1.936
0.105
0.713
1.694
1.965
0.060
0.473
1.512
1.937
0.588
1.613
1.953
1.995
0.064
0.494
1.533
1.941
0.105
0.715
1.695
1.965
0.280
1.239
1.884
1.988
0.133
0.833
1.754
1.972
0.206
103
Tabulated Reflection and Transmission Coefficients
Graphite! Epoxy
Graphite/ Epoxy
Graphite/ Epoxy
Graphite/ Epoxy
Graphite/ Epoxy
Graphite/ Epoxy
Graphite/ Epoxy
Graphite/ Epoxy
Graphite/ Epoxy
Graphite/ Epoxy
Graphite/ Epoxy
Graphite/ Epoxy
Graphite/ Epoxy
Graphite/ Epoxy
Graphite/ Epoxy
Graphite/ Epoxy
Graphite/ Epoxy
Graphite/ Epoxy
Graphite/ Epoxy
Graphite/ Epoxy
Graphite/ Epoxy
Graphite/ Epoxy
Graphite/ Epoxy
Graphite/ Epoxy
Graphite/ Epoxy
Graphite/ Epoxy
Graphite/ Epoxy
Nickel Steel
Nickel Steel
Nickel Steel
Nickel Steel
Nickel Steel
Nickel Steel
Nickel Steel
Nickel Steel
Nickel Steel
Nickel Steel
Nickel Steel
Nickel Steel
Nickel Steel
Nickel Steel
Nickel Steel
Nickel Steel
Nickel Steel
Nickel Steel
Nickel Steel
Nickel Steel
Nickel Steel
Nickel Steel
Nickel Steel
Nickel Steel
Nickel Steel
Nickel Steel
Nickel Steel
Nickel Steel
Nickel Steel
Nickel Steel
Nickel Steel
Nickel Steel
Nickel Steel
Nickel Steel
Nickel Steel
Nickel Steel
Boron/ Epoxy
Boron/ Epoxy
Boron/ Epoxy
Cast Iron Alloys
Cast Iron Alloys
Cast Iron Alloys
Cast Iron Alloys
Glass (Fused Silica)
Glass (Fused Silica)
Glass (Fused Silica)
Glass (Fused Silica)
Nickel Steel
Nickel Steel
Nickel Steel
Nickel Steel
Polymethyl Methacrylate (PMMA)
Polymethyl Methacrylate (PMMA)
Polymethyl Methacrylate (PM MA)
Polymethyl Methacrylate (PMMA)
Stainless Steel
Stainless Steel
Stainless Steel
Stainless Steel
Titanium
Titanium
Titanium
Titanium
Aluminum Alloys
Aluminum Alloys
Aluminum Alloys
Aluminum Alloys
Beryllium Copper
Beryllium Copper
Beryllium Copper
Beryllium Copper
Boron/ Epoxy
Boron/ Epoxy
Boron/ Epoxy
Boron/ Epoxy
Cast Iron Alloys
Cast Iron Alloys
Cast Iron Alloys
Cast Iron Alloys
Glass (Fused Silica)
Glass (Fused Silica)
Glass (Fused Silica)
Glass (Fused Silica)
Graphite/ Epoxy
Graphite/ Epoxy
Graphite/ Epoxy
Graphite/ Epoxy
Polymethyl Methacrylate (PMMA)
Polymethyl Methacrylate (PMMA)
Polymethyl Methacrylate (PMMA)
Polymethyl Methacrylate (PMMA)
Stainless Steel
Stainless Steel
Stainless Steel
Stainless Steel
Titanium
Titanium
Titanium
Titanium
9.10
9.10
9.10
9.10
9.10
9.10
9.10
9.10
9.10
9.10
9.10
9.10
9.10
9.10
9.10
9.10
9.10
9.10
9.10
9.10
9.10
9.10
9.10
9.10
9.10
9.10
9.10
4.91
4.91
4.91
4.91
4.91
4.91
4.91
4.91
4.91
4.91
4.91
4.91
4.91
4.91
4.91
4.91
4.91
4.91
4.91
4.91
4.91
4.91
4.91
4.91
4.91
4.91
4.91
4.91
4.91
4.91
4.91
4.91
4.91
4.91
4.91
4.91
9.87
9.87
9.87
5.73
5.73
5.73
5.73
5.72
5.72
5.72
5.72
4.91
4.91
4.91
4.91
2.52
2.52
2.52
2.52
4.96
4.96
4.96
4.96
5.03
5.03
5.03
5.03
5.04
5.04
5.04
5.04
3.90
3.90
3.90
3.90
9.87
9.87
9.87
9.87
5.73
5.73
5.73
5.73
5.72
5.72
5.72
5.72
9.10
9.10
9.10
9.10
2.52
2.52
2.52
2.52
4.96
4.96
4.96
4.96
5.03
5.03
5.03
5.03
Grouped by Material Pair
206.8
206.8
206.8
206.8
206.8
206.8
206.8
206.8
206.8
206.8
206.8
206.8
206.8
206.8
206.8
206.8
206.8
206.8
206.8
206.8
206.8
206.8
206.8
206.8
206.8
206.8
206.8
199.9
199.9
199.9
199.9
199.9
199.9
199.9
199.9
199.9
199.9
199.9
199.9
199.9
199.9
199.9
199.9
199.9
199.9
199.9
199.9
199.9
199.9
199.9
199.9
199.9
199.9
199.9
199.9
199.9
199.9
199.9
199.9
199.9
199.9
199.9
199.9
195.0
195.0
195.0
240.0
240.0
240.0
240.0
72.0
72.0
72.0
72.0
199.9
199.9
199.9
199.9
7.6
7.6
7.6
7.6
190.3
190.3
190.3
190.3
113.8
113.8
113.8
113.8
70.3
70.3
70.3
70.3
124.1
124.1
124.1
124.1
195.0
195.0
195.0
195.0
240.0
240.0
240.0
240.0
72.0
72.0
72.0
72.0
206.8
206.8
206.8
206.8
7.6
7.6
7.6
7.6
190.3
190.3
190.3
190.3
113.8
113.8
113.8
113.8
1.0
0.070
1.070
10.0
100.0
0.1
1.0
10.0
100.0
0.1
1.0
10.0
100.0
0.1
1.0
10.0
100.0
0.1
1.0
10.0
100.0
0.1
1.0
10.0
100.0
0.1
1.0
10.0
100.0
0.1
1.0
10.0
100.0
0.1
1.0
10.0
100.0
0.1
1.0
10.0
100.0
0.1
1.0
10.0
100.0
0.1
1.0
10.0
100.0
0.1
1.0
10.0
100.0
0.1
1.0
10.0
100.0
0.1
1.0
10.0
100.0
0.1
1.0
10.0
100.0
0.840
0.983
-0.897
-0.296
0.689
0.964
-0.694
0.287
0.895
0.989
-0.894
-0.283
0.696
0.965
-0.141
0.765
0.974
0.997
-0.888
-0.256
0.711
0.967
-0.817
0.002
0.819
0.980
-0.548
0.489
0.934
0.993
-0.773
0.122
0.855
0.984
-0.658
0.347
0.907
0.990
-0.823
-0.014
0.814
0.980
-0.511
0.527
0.940
0.994
-0.696
0.283
0.894
0.989
0.148
0.862
0.985
0.999
-0.808
0.030
0.828
0.981
-0.695
0.285
0.895
0.989
1.840
1.983
0.103
0.704
1.689
1.964
0.306
1.287
1.895
1.989
0.106
0.717
1.696
1.965
0.859
1.765
1.974
1.997
0.112
0.744
1.711
1.967
0.183
1.002
1.819
1.980
0.452
1.489
1.934
1.993
0.227
1.122
1.855
1.984
0.342
1.347
1.907
1.990
0.177
0.986
1.814
1.980
0.489
1.527
1.940
1.994
0.304
1.283
1.894
1.989
1.148
1.862
1.985
1.999
0.192
1.030
1.828
1.981
0.305
1.285
1.895
1.989
104
Tabulated Reflection and Transmission Coefficients
A7777
Polymethyl Methacrylate (PMMA)
Polymethyl Methacrylate (PMMA)
Polymethyl Methacrylate (PMMA)
Polymethyl Methacrylate (PMMA)
Polymethyl Methacrylate (PMMA)
Polymethyl Methacrylate (PMMA)
Polymethyl Methacrylate (PMMA)
Polymethyl Methacrylate (PMMA)
Polymethyl Methacrylate (PMMA)
Polymethyl Methacrylate (PMMA)
Polymethyl Methacrylate (PMMA)
Polymethyl Methacrylate (PMMA)
Polymethyl Methacrylate (PMMA)
Polymethyl Methacrylate (PMMA)
Polymethyl Methacrylate (PMMA)
Polymethyl Methacrylate (PMMA)
Polymethyl Methacrylate (PMMA)
Polymethyl Methacrylate (PMMA)
Polymethyl Methacrylate (PMMA)
Polymethyl Methacrylate (PMMA)
Polymethyl Methacrylate (PMMA)
Polymethyl Methacrylate (PMMA)
Polymethyl Methacrylate (PMMA)
Polymethyl Methacrylate (PMMA)
Polymethyl Methacrylate (PMMA)
Polymethyl Methacrylate (PMMA)
Polymethyl Methacrylate (PM MA)
Polymethyl Methacrylate (PMMA)
Polymethyl Methacrylate (PMMA)
Polymethyl Methacrylate (PMMA)
Polymethyl Methacrylate (PMMA)
Polymethyl Methacrylate (PMMA)
Polymethyl Methacrylate (PMMA)
Polymethyl Methacrylate (PMMA)
Polymethyl Methacrylate (PMMA)
Polymethyl Methacrylate (PMMA)
Stainless Steel
Stainless Steel
Stainless Steel
Stainless Steel
Stainless Steel
Stainless Steel
Stainless Steel
Stainless Steel
Stainless Steel
Stainless Steel
Stainless Steel
Stainless Steel
Stainless Steel
Stainless Steel
Stainless Steel
Stainless Steel
Stainless Steel
Stainless Steel
Stainless Steel
Stainless Steel
Stainless Steel
Stainless Steel
Stainless Steel
Stainless Steel
Stainless Steel
Stainless Steel
Stainless Steel
Aluminum Alloys
Aluminum Alloys
Aluminum Alloys
Aluminum Alloys
Beryllium Copper
Beryllium Copper
Beryllium Copper
Beryllium Copper
Boron/ Epoxy
Boron/ Epoxy
Boron/ Epoxy
Boron/ Epoxy
Cast Iron Alloys
Cast Iron Alloys
Cast Iron Alloys
Cast Iron Alloys
Glass (Fused Silica)
Glass (Fused Silica)
Glass (Fused Silica)
Glass (Fused Silica)
Graphite/ Epoxy
Graphite/ Epoxy
Graphite/ Epoxy
Graphite/ Epoxy
Nickel Steel
Nickel Steel
Nickel Steel
Nickel Steel
Stainless Steel
Stainless Steel
Stainless Steel
Stainless Steel
Titanium
Titanium
Titanium
Titanium
Aluminum Alloys
Aluminum Alloys
Aluminum Alloys
Aluminum Alloys
Beryllium Copper
Beryllium Copper
Beryllium Copper
Beryllium Copper
Boron/ Epoxy
Boron/ Epoxy
Boron/ Epoxy
Boron/ Epoxy
Cast Iron Alloys
Cast Iron Alloys
Cast Iron Alloys
Cast Iron Alloys
Glass (Fused Silica)
Glass (Fused Silica)
Glass (Fused Silica)
Glass (Fused Silica)
Graphite/ Epoxy
Graphite/ Epoxy
Graphite/ Epoxy
Graphite/ Epoxy
Nickel Steel
Nickel Steel
Nickel Steel
2.52
2.52
2.52
2.52
2.52
2.52
2.52
2.52
2.52
2.52
2.52
2.52
2.52
2.52
2.52
2.52
2.52
2.52
2.52
2.52
2.52
2.52
2.52
2.52
2.52
2.52
2.52
2.52
2.52
2.52
2.52
2.52
2.52
2.52
2.52
2.52
4.96
4.96
4.96
4.96
4.96
4.96
4.96
4.96
4.96
4.96
4.96
4.96
4.96
4.96
4.96
4.96
4.96
4.96
4.96
4.96
4.96
4.96
4.96
4.96
4.96
4.96
4.96
5.04
5.04
5.04
5.04
3.90
3.90
3.90
3.90
9.87
9.87
9.87
9.87
5.73
5.73
5.73
5.73
5.72
5.72
5.72
5.72
9.10
9.10
9.10
9.10
4.91
4.91
4.91
4.91
4.96
4.96
4.96
4.96
5.03
5.03
5.03
5.03
5.04
5.04
5.04
5.04
3.90
3.90
3.90
3.90
9.87
9.87
9.87
9.87
5.73
5.73
5.73
5.73
5.72
5.72
5.72
5.72
9.10
9.10
9.10
9.10
4.91
4.91
4.91
Grouped by Material Pair
7.6
7.6
7.6
7.6
7.6
7.6
7.6
7.6
7.6
7.6
7.6
7.6
7.6
7.6
7.6
7.6
7.6
7.6
7.6
7.6
7.6
7.6
7.6
7.6
7.6
7.6
7.6
7.6
7.6
7.6
7.6
7.6
7.6
7.6
7.6
7.6
190.3
190.3
190.3
190.3
190.3
190.3
190.3
190.3
190.3
190.3
190.3
190.3
190.3
190.3
190.3
190.3
190.3
190.3
190.3
190.3
190.3
190.3
190.3
190.3
190.3
190.3
190.3
70.3
70.3
70.3
70.3
124.1
124.1
124.1
124.1
195.0
195.0
195.0
195.0
240.0
240.0
240.0
240.0
72.0
72.0
72.0
72.0
206.8
206.8
206.8
206.8
199.9
199.9
199.9
199.9
190.3
190.3
190.3
190.3
113.8
113.8
113.8
113.8
70.3
70.3
70.3
70.3
124.1
124.1
124.1
124.1
195.0
195.0
195.0
195.0
240.0
240.0
240.0
240.0
72.0
72.0
72.0
72.0
206.8
206.8
206.8
206.8
199.9
199.9
199.9
0.1
1.0
10.0
100.0
0.1
1.0
10.0
100.0
0.1
1.0
10.0
100.0
0.1
1.0
10.0
100.0
0.1
1.0
10.0
100.0
0.1
1.0
10.0
100.0
0.1
1.0
10.0
100.0
0.1
1.0
10.0
100.0
0.1
1.0
10.0
100.0
0.1
1.0
10.0
100.0
0.1
1.0
10.0
100.0
0.1
1.0
10.0
100.0
0.1
1.0
10.0
100.0
0.1
1.0
10.0
100.0
0.1
1.0
10.0
100.0
0.1
1.0
10.0
-0.958
-0.645
0.367
0.912
-0.981
-0.827
-0.027
0.809
-0.970
-0.735
0.209
0.877
-0.986
-0.866
-0.162
0.756
-0.953
-0.614
0.411
0.920
-0.974
-0.766
0.140
0.860
-0.985
-0.862
-0.148
0.762
-0.984
-0.854
-0.120
0.774
-0.974
-0.765
0.143
0.860
-0.569
0.467
0.930
0.993
-0.785
0.093
0.847
0.984
-0.675
0.320
0.902
0.990
-0.832
-0.044
0.803
0.978
-0.533
0.506
0.936
0.993
-0.711
0.256
0.888
0.988
-0.828
-0.029
0.808
0.042
0.355
1.367
1.912
0.019
0.173
0.973
1.809
0.030
0.265
1.209
1.877
0.014
0.134
0.838
1.756
0.047
0.386
1.411
1.920
0.026
0.234
1.140
1.860
0.015
0.138
0.852
1.762
0.016
0.146
0.880
1.774
0.026
0.235
1.143
1.860
0.431
1.467
1.930
1.993
0.215
1.093
1.847
1.984
0.325
1.320
1.902
1.990
0.168
0.956
1.803
1.978
0.467
1.506
1.936
1.993
0.289
1.256
1.888
1.988
0.172
0.971
1.808
105
Tabulated Reflection and Transmission Coefficients
Stainless Steel
Stainless Steel
Stainless Steel
Stainless Steel
Stainless Steel
Stainless Steel
Stainless Steel
Stainless Steel
Stainless Steel
Titanium
Titanium
Titanium
Titanium
Titanium
Titanium
Titanium
Titanium
Titanium
Titanium
Titanium
Titanium
Titanium
Titanium
Titanium
Titanium
Titanium
Titanium
Titanium
Titanium
Titanium
Titanium
Titanium
Titanium
Titanium
Titanium
Titanium
Titanium
Titanium
Titanium
Titanium
Titanium
Titanium
Titanium
Titanium
Titanium
Nickel Steel
Polymethyl Methacrylate (PMMA)
Polymethyl Methacrylate (PMMA)
Polymethyl Methacrylate (PMMA)
Polymethyl Methacrylate (PMMA)
Titanium
Titanium
Titanium
Titanium
Aluminum Alloys
Aluminum Alloys
Aluminum Alloys
Aluminum Alloys
Beryllium Copper
Beryllium Copper
Beryllium Copper
Beryllium Copper
Boron/ Epoxy
Boron/ Epoxy
Boron/ Epoxy
Boron/ Epoxy
Cast Iron Alloys
Cast Iron Alloys
Cast Iron Alloys
Cast Iron Alloys
Glass (Fused Silica)
Glass (Fused Silica)
Glass (Fused Silica)
Glass (Fused Silica)
Graphite/ Epoxy
Graphite/ Epoxy
Graphite/ Epoxy
Graphite/ Epoxy
Nickel Steel
Nickel Steel
Nickel Steel
Nickel Steel
Polymethyl Methacrylate (PMMA)
Polymethyl Methacrylate (PMMA)
Polymethyl Methacrylate (PMMA)
Polymethyl Methacrylate (PMMA)
Stainless Steel
Stainless Steel
Stainless Steel
Stainless Steel
4.96
4.96
4,96
4.96
4.96
4.96
4.96
4.96
4.96
5.03
5.03
5.03
5.03
5.03
5.03
5.03
5.03
5.03
5.03
5.03
5.03
5.03
5.03
5.03
5.03
5.03
5.03
5.03
5.03
5.03
5.03
5.03
5.03
5.03
5.03
5.03
5.03
5.03
5.03
5.03
5.03
5.03
5.03
5.03
5.03
4.91
2.52
2.52
2.52
2.52
5.03
5.03
5.03
5.03
5.04
5.04
5.04
5.04
3.90
3.90
3.90
3.90
9.87
9.87
9.87
9.87
5.73
5.73
5.73
5.73
5.72
5.72
5.72
5.72
9.10
9.10
9.10
9.10
4.91
4.91
4.91
4.91
2.52
2.52
2.52
2.52
4.96
4.96
4.96
4.96
Grouped by Material Pair
190.3
190.3
190.3
190.3
190.3
190.3
190.3
190.3
190.3
113.8
113.8
113.8
113.8
113.8
113.8
113.8
113.8
113.8
113.8
113.8
113.8
113.8
113.8
113.8
113.8
113.8
113.8
113.8
113.8
113.8
113.8
113.8
113.8
113.8
113.8
113.8
113.8
113.8
113.8
113.8
113.8
113.8
113.8
113.8
113.8
199.9
7.6
7.6
7.6
7.6
113.8
113.8
113.8
113.8
70.3
70.3
70.3
70.3
124.1
124.1
124.1
124.1
195.0
195.0
195.0
195.0
240.0
240.0
240.0
240.0
72.0
72.0
72.0
72.0
206.8
206.8
206.8
206.8
199.9
199.9
199.9
199.9
7.6
7.6
7.6
7.6
190.3
190.3
190.3
190.3
100.0
0.1
1.0
10.0
100.0
0.1
1.0
10.0
100.0
0.1
1.0
10.0
100.0
0.1
1.0
10.0
100.0
0.1
1.0
10.0
100.0
0.1
1.0
10.0
100.0
0.1
1.0
10.0
100.0
0.1
1.0
10.0
100.0
0.1
1.0
10.0
100.0
0.1
1.0
10.0
100.0
0.1
1.0
10.0
100.0
0.979
0.119
0.854
0.984
0.998
-0.710
0.258
0.889
0.988
-0.721
0.237
0.884
0.988
-0.867
-0.169
0.753
0.972
-0.794
0.068
0.839
0.983
-0.897
-0.298
0.688
0.964
-0.695
0.285
0.895
0.989
-0.819
-0.002
0.817
0.980
-0.895
-0.285
0.695
0.965
-0.143
0.764
0.974
0.997
-0.889
-0.258
0.710
0.967
1.979
1.119
1.854
1.984
1.998
0.290
1.258
1.889
1.988
0.279
1.237
1.884
1.988
0.133
0.831
1.753
1.972
0.206
1.068
1.839
1.983
0.103
0.702
1.688
1.964
0.305
1.285
1.895
1.989
0.181
0.998
1.817
1.980
0.105
0.715
1.695
1.965
0.857
1.764
1.974
1.997
0.111
0.742
1.710
1.967
106
Appendix D: Visual Representations of Reflections and Transmissions at
Selected Material Interfaces
The following figures are visual representations of the reflections and transmissions of a
normalized incident wave pulse at the interface between selected material pairs which make up a
composite rod. The selected pairs were chosen because the drastic visual transformations of the
transmitted and reflected wave pulses highlight some of the interesting effects of boundary interactions
and the importance of the sequence of the materials on the resultant wave pulses.
Material A
Figure D 1: Composite rod with internal interface.
All of the reflection/ transmission pairs are developed from a normalized incident pulse with
amplitude and pulse width of unity shown in Figure 41. The resultant pulses of both relevant transmissions
(Material A 4
Material B and Material B 4 Material A) are shown on the same axes to highlight the
dependence of the wave pulse transformations on the direction of the transmission. In each case, the
reflected waves will be identical inversions of one another because R ,A-B = -R
BA.
The distance of the
transmitted waves from the vertical axis (ie. the interface) is dependent upon the wave speed of the
transmitted material, so some of the transmitted waves will travel noticeably further from the interface
in the same time step.
For reference, each figure is accompanied by the corresponding reflection and transmission
coefficients for both directional wave transmissions.
107
A,= A2
Pulse Amplitude
)
-1
ci.
N
<-- w=1-
\j/
Figure D 2: Normalized incident wave pulse.
108
MATERIAL B (E 2 ,c 2
)
A
MATERIAL A (Ei,c
Reflected and Transmitted Pulses at a Boundary
Material A: Boron/ Epoxy; Material B: Nickel Steel
-
Material
Material
Material
Material
A
A
B
B
Material B: Reflection
Matenal B: Transmission
A: Reflection
Material A: Transmission
nMaterial
>
1.5
1
A
-5
-4
-3
2
2
3
4
5
-0.5
Boron/ Epoxy
Nickel Steel
c = 9.87 km/
c = 4.91 km/
E = 195.0 GN/m 2
E = 199.9 GN/m 2
Reflection Coefficients
-0.346
R
RBA = 0.346
Transmission Coefficients
= 0.654
TB-A =
1.347
The transmission across an interface between boron/ epoxy and nickel steel provides a visual
representation of the effects of transmission between two materials with similar moduli of elasticity but
significantly different primary wave speeds. The near equivalent elastic moduli between the two materials
results in the majority of the pulse's energy being transmitted through the boundary because the elastic
modulus and cross-sectional area terms approximately cancel in the reflection coefficient expression, and
the reflection coefficient is determined nearly exclusively by the difference between the two primary
wave speeds. Since the primary wave speed of boron/epoxy is nearly twice that of nickel steel, the
reflection coefficient reduces to approximately Rk
~.
A greater difference between the two
wave speeds leads to less of the wave pulse's total energy being transmitted through the interface and
more being reflected.
The higher wave speed of boron/ epoxy results in the amplification of the pulse transmission into
this medium, both in the pulse width and its amplitude, and the distance the pulse travels from the
interface in the same time step.
109
Reflected and Transmitted Pulses at a Boundary
Material A: Cast Iron; Material B: Glass
2
F-
Material A -> Material
Material
Material A
Material B-> Material
- - -- - --B -,Material
-- - - - -Material
-
--
1.5
d
-
B:
B
A:
A:
Reflection
Transmission
Reflection
Transmission
I
/
0.5
I,
I
-5
-4
I
-3
2
a
-1
2
I
I
I
3
4
5
-0.5
Cast Iron
Glass
Reflection Coefficients
c = 5.73 km/s
c = 5.72 km/s
R kA-B = 0.538
E = 240.0 GN/m 2
E = 72.0 GN/m 2
R
B--A
=
0.538
Transmission Coefficients
TkA-B =
T
1.538
=B-+A
0.462
The almost equivalent primary wave speeds of glass and cast iron mean that the transmitted
waves will travel equivalent distances in equal time steps and maintain the original pulse width, so the
transmitted wave pulses overlap. However, the drastically different moduli lead to significant changes in
the amplitude of the transmitted waves. As one might expect, wave energy transmitted into the far more
elastic glass medium results in large displacements of elemental particles, yielding a large wave amplitude.
The same wave energy yields much smaller amplitudes of motion in the more rigid cast iron.
110
Reflected and Transmitted Pulses at a Boundary
Material A: Aluminum Alloy; Material B: Glass
-
Material A
->
-----
Material A
Material B
Material B
->
-----
1.5
*
3
-2
-3
-2
Reflection
Transmission
Reflection
Transmission
I
1
0
-
I
-4
B:
B:
A:
A:
,,
I,,,
I
0.5
-5
->
Material
Material
Material
Material
/ 4I
1
I
->
1
2
I
I
3
4
I
-0.5
Aluminum Alloy
Glass
Reflection Coefficients
c = 5.04 km/s
c = 5.72 km/s
RgA-B = 0.052
E = 70.3 GN/m 2
E = 72.0 GN/m 2
R
BA,
=
-0.052
Transmission Coefficients
= 1.051
TkB-A =
0.948
Nearly equivalent moduli of elasticity and similar, if not equal, primary wave speeds make this
material pair interesting because they almost approach a continuous, homogenous rod with no internal
interface, so very little of the wave pulse's energy is reflected and both of the transmitted pulses maintain
an amplitude very near to that of the incident pulse.
111
Reflected and Transmitted Pulses at a Boundary
Material A: Polymethyl Methacrylate (PMMA)
Material B: Titanium
2
-
AA B
B-
Material B:
Material B:
Mate al A:
Material A:
Reflection
Transmission
Reflection
Transmission
-
1,5
Material
- Material
Material
Material
-
/
-
0.5
- - - - --
i
i
I
I
I---I
i
I
-5
-4
-3
2
3
-1
1
2
3
4
-0 5
PMMA
Titanium
Reflection Coefficients
c = 2.52 km/s
c = 5.03 km/
RgA+B = -0.765
E = 7.6 GN/m 2
E = 113.8 GN/m 2
R
B-A
=
0.765
Transmission Coefficients
TgA_,
= 0.235
TB-A =
1.764
The material properties of PMMA result in interesting wave pulse transmissions with any of the
more rigid common materials considered. Its extreme flexibility compared to that of titanium lead to large
displacements when the pulse's energy is received, and the pulse transmission into PMMA shows a large
amplification of pulse amplitude. The relatively low wave speed leads to the spatial contraction of the
pulse width when a wave pulse is transmitted into PMMA coming from many common materials, and the
pulse will not travel as far from the interface in a given time step because it moves relatively slow.
112
Appendix E: Expanding the Wave Function Arguments
E.1 Problem Definition
In considering the wave behavior in a composite body consisting of two rods (Fig El), two
interfacial boundary conditions must be considered at the junction between the two media. The first is a
compatibility of the displacement, which requires that the two rod segments remain in contact as the
wave traverses the interface. As such, the displacement due to wave propagation of an infinitesimal
element at the boundary must be the same when considered from either side. The second condition is an
equality of force across the boundary. Namely, the force experienced by an infinitesimal element located
at the boundary must be the same when considered from either side, otherwise there would be a step
jump in force and an infinitesimal element would experience infinite acceleration across the boundary.
A.-I *W
WA
V,
X
x 0
Figure E 1: Composite rod with junction.
Using the principle of superposition, these boundary conditions are represented mathematically
as:
Condition 1:
Condition 2:
i (0, c 1 t) + 4 r (0, Cit) =
E1A1
+
aft(O,cit)
OX
(Eq El)
(0, c2 t)
E1A1 'fr(o,cit)
ax
= EfO,cit)
ax
+
aG(o,c2 t)
a=E2A2
ax
x
Eq E2)
Here, the subscripts i, r, and t represent the incident, reflected, and transmitted wave pulses, respectively.
113
The following convention is adopted:
i(x, cit)
=
cit)
tr(X,
fh(xi, c1 t)
= gr(Xr,
S(Xc 2 t)
=
c t)
ft(xt,cz t)
(Eq E3a)
f (Xi - ct)
=
=
=
gr(xr + ct)
ft(xt
(Eq E3b)
(Eq E3c)
- c2 t)
where cp is the primary (or longitudinal) wave speed of the particular medium of propagation, and the
arbitrary wave functions
f(x,cpt) and g(xcpt) represent forward and backward propagating waves,
respectively.
E.2 Deriving the Reflection and Transmission Coefficients
Applying the chain rule, the partial derivatives of Condition 2 (Eq E2) can be expressed as follows:
df(x-c t)
_Of(x-cpt)
a(x-cpt)
_af(x-cpt)
a(x-c t)
Ox
ax
a(x-cyt)
a(x-cyt)
ax
_
ag(x+cyt) a(x+cyt)
Ox
ax
_
a(x+cyt)
-
ag(x+cyt) a(x+cyt)
Ox
a(x+cyt)
_
(x +
g
ct)
C t)
+
ag(x+cyt)
_
Thus, adapting the convention of Eqs E3 (a - c) and specifying the conditions at the interface
(x = 0), Eqs El and E2 become:
f (-c 1 t) + gr (ct)
(Eq E4)
= ft (-c 2 t)
E 1A 1[f'j(-cjt) + g',(cit)] = E2 A 2 f't(-c 2 t)
(Eq E5)
Because Eq E4 is written in terms of the wave functions fi, ft, and gr, and Eq E5 in terms of their
respective derivatives, Eq E5 must be fully integrated. Note the following integral substitutions:
l1
Ar
(ct)
f4 EAig'r (ct)dt = E1 A 1 fog'r(cit)
f
E2 A 2f't
(-c2 t)dt
= E2 A 2
f
t)
d(ci
Cl
t(-czt)
-oC2
=
=
C,
E A1
C'
f (-c1 t)
I
d(-c2 t) = - Eft(-C
C
2
2
)
t
t
f0 EAf'j (-cit)dt = E1 A 1 ff'j(-cit) -d(-clt)
Cl
Thus, the integration of Eq E5 yields:
E1A1
[-fi(-C1 t) + gr (Ct
Cl
=
E2A2
ft (-C2t)
(Eq E6)
C2
Now, Eqs E4 and E6 form a system of equations that can be solved using elimination to yield the
following expressions for the reflected and transmitted wave functions in terms of the incident wave
114
function. The coefficients in these expressions are aptly named the reflection coefficient and the
transmission coefficient, respectively.
g' (r10
(r
(c- t) =
= (E 1 AlC 2 E2 A 2 C1
(-C2) =
E1 A c+A
1
2c
fi(- Ct)
(Eq E7)
(q
fi( ct)
7
(Eq E8)
E.3 Expanding the Wave Function Arguments
Eqs E7 and E8 are defined only at the boundary, x
=
0, and the goal here is to expand the
arguments to include the entire domains of both rod segments. To expand the definition of the function
arguments to include the entire range of x, the following thought process and mathematical manipulation
are useful.[6]
The central point regarding the definitions in Eqs E7 and E8 is that, although the expressions are
specifically defined at the junction x = 0 for time t, these expressions actually define the relationships
between these wave functions at any time-position pairing when the wave function arguments are
equivalent to one another. There is nothing special about the junction. To illustrate this, the arguments in
Eqs E7 and E8 can be replaced with a dummy variable which also signifies that the arguments of the
various waveforms are equivalent to one another:
Yr (T) = Rfi(T)
(Eq E9)
ft()
(Eq ElO)
= Tfi()
Although the phase value, r, is equivalent for all three wave functions, its representation in terms
of the particularx locations of the wave pulses at a given time t will be different. To derive the appropriate
arguments of each waveform, first recall the previously defined phase values for each waveform, namely
the respective arguments of the arbitrary wave functions of Eqs E3 (a - c), now given the designations ri,
rr, and rt for clarity. To account for the physical changes that occur when a wave is transmitted into a new,
nondispersive medium and/or reflected at a boundary (specifically that the frequency does not change
but the spatial characteristics [i.e. amplitude and/or wavelength] can, depending on the respective wave
speeds of the two dissimilar media) it is appropriate to associate the change resulting from the boundary
interaction(s) with the spatial element as opposed to the temporal one. Hence these arguments are
rewritten as follows:
115
-
Incident Waveform:
T'i
Reflected Waveform:
T'r -
Transmitted Waveform:
T't -
- - ti -1i
(Eq Ella)
xr
(Eq Elib)
C1
C1
- tr +
C1
C1
C2
-tt-xt
(Eq Elic)
C2
where (xk, tk) are the particular time-space pairs at which each individual waveform is considered. In
practice, it is most useful to consider all of the waveforms at the same point in time, so let ti = tr = tt = t.
Next, satisfying Eqs E9 and E10 requires that the arguments of all three waveforms be equivalent to one
another.
Equating Eq Ella and Eq Elib:
= t+
t - Xi
C1
C1
Xr-Xi
= -Xr
(Eq E12)
Similarly, equating Eq Ella and Eq Ellc:
t-
C1
xi = t -
-xt
C
2
xi =*xt
(Eq E13)
C2
Thus, the arguments of Eqs E7 and E8 can be expanded to account for all values of x at a given time t, and
the equations are rewritten in their general forms.
gr[xr(t),cit] = Rkf[-xj(t),cjt]
(Eq E14)
ft[xt(t), c2 t] = Tkf
(Eq E15)
Xi
(t), c2 t]
Here, the spatial element is modified slightly to include its time "dependence", for reasons that will
become more apparent through the examples included in the following section."
E.4 Examples
To better illustrate the meaning and application of Eqs E14 and E15, several example calculations
are included. First, consider an incident wave consisting of two discrete pulses, modeled mathematically
with the Dirac delta function.
The position of a wave (or, more precisely, of the particles perturbed by the wave's passing) is not necessarily
explicitly time dependent, although it potentially could be. More specifically, where a particle or particles will be
observed depends on when they are observed. Since the position varies in time, it can (and should) be specified with
respect to a particular time, t.
16
116
i,
fi[xi(t), c1 t] = 8[xil(t) - c 1t + 2] + 6[(xi 2 (t) - c 1 t + 4)]
As defined, xil(O) = -2 and xi 2 (0)
(Eq E16)
-4. For this example, let cl = 1 and c2 = 2, with arbitrary units of
=
unit distance . Thus, at time t = 0, the incident wave is given by
the following sketch:
time step
Amplitude
t=o
A
C'=1
- 1
I
I
-I
-
-4
-6
-5
-4
I
-3
I
-2
-1
1
2
3
4
5
6
Figure E 2: Discrete incident waveform.
The incident waveform will first encounter the boundary, x = 0, at time step t = 2, and the
waveform will be fully transmitted/reflected two time steps later at t = 4. As such, at t = 5 only the
resultant waveforms will remain. However, were the incident wave pulse able to continue its travel
through the boundary uninterrupted and at the same speed of propagation, cl = 1, its position at time
step t = 5 would be given by Figure E3.
Amplitude
t=5
A
- 1
-6
II
I
I
II
II
I
II
I
-5
-4
-3
-2
I
I
-1
T
c'=1
I
1
I
2
1
3
4
5
6
Figure E 3: Projected future position of discrete incident pulse in absence of boundary at x = 0.
Using the projected future position of the incident pulse at time step t = 5, a summary of all required
information to fully apply Eqs E14 and E15, as well as the corresponding results, is provided in Table El.
117
1
units
time step
2 units
time step
Cj
C2
1 unit
3 units
xii(5)
Xi2(5)
-1 unit
Xrl(5) = -ii(5)
Xr2(5) =
xti(5)
=
-xi 2(5)
-3 units
2 units
cx ii(5)
C1
6 units
Xt2(5) = -Xi2(5)
C1
Rk= Tk
2
Table E 1: Required values to solve discrete example.
Thus, the reflected and transmitted waveforms can be easily placed using Eqs E14 and E15. The
resultant waveforms are sketched below.
t=5
Amplitude
A
- 1
-6
I
-5
II
-4
Ii
-3
C'=1
-
c,=2
0.5
I7-2
-1
X
\11
1
2
3
4
5
6
Figure E 4: Resultant waveforms (discrete example).
To fully illustrate that these are, indeed, the correct locations of the reflected and transmitted
waveforms, the individual time steps of the reflection and transmission processes are shown below. For
clarity, the reflected and transmitted waveforms are isolated in their respective sketches.
118
t=o
Amplitude
-1
I
I
-I
|
-Uz -61
-1
-2
-3
-4
-5
i
1
i
i
2
I
I
I
3
4
5
6
t=2
Amplitude
S-5 -4
.3
-2
-4
-3
-2
-5
-6
1
-
1
\/
-1
2
3
4
5
6
2
3
4
5
6
t=2
Amplitude
1
-1
-2
-3
-4
-5
-6
--V X
6
5
4
3
2
t=3
Amplitude
- -0.5
-1
-2
-3
-4
-5
-6
---2
(1
\
>
4
3
x
6
5
t=4
Amplitude
--1
- -0.5
-6
-5
-4
-3
-2
1
-1
2
3
4
5
6
t=5
Amplitude
-0.5
x
-6
-5
-4
-3
-2
1
-1
Figure E 5: Discrete waveform reflection in individual time steps.
119
2
3
4
5
6
t=o
Amplitude
H
I
~
-6
-4
-5
- 1
-2
-3
I
I
-1
1
I
I
2
3
I
I
4
5
6
t=1
Amplitude
x
-4
-5
-6
5
4
3
2
\1
-1
-2
-3
6
t=2
Amplitude
<:
'7L
-1
-~
-6
-1
-2
-3
-4
-5
/
6
5
4
2
1
\/
\
t=3
Amplitude
---
-2
-3
-4
-5
-6
1
-1
-60.
3
2
1
c,=2
Amplitude
5
4
L
6
4
5
4
5
t=4
/ .\
x
-6
-5
-
3
-
1
\/
1
2
3
t=5
Amplitude
-
6
c=
-0.5
x
-6
-5
-4
-3
-2
1
-1
Figure E 6: Discrete waveform transmission in individual time steps.
120
2
3
4
5
6
It should now be apparent from the previous example that the value of x;(t) in Eqs E14 and E15
refers to the projected location of the incident wave at the time step of interest, were it allowed to
continue its trajectory unimpeded by any boundaries and at its same speed of propagation.
A second example, a continuous step function, is provided below. For this example, let ci = 2,
c2 =
, as before. Here, the waveforms will be identified by the two endpoints of the
1, and let R = Tf=
step function, given the designations x,1(t) and x12(t) as in the previous example. All of the values required
to evaluate Eqs E14 and E15, and the corresponding results, are summarized in Table E2.
t=o
Amplitude
c =2
-1
I
-6
-5
I
I
I
I
I
I
-4
-3
I
I
I
I
I
I
-2
-1
2
1
V
3
4
5
6
Figure E 7: Continuous incident step function waveform.
Amplitude
t=4
A
c =2
- 1
.eI -II I I I I I
-6
-5
I
I
I
-4
-3
-2
I
-1
I
1
V
I
2
I
3
I
I
I
I
I
I
I
I
I
4
I
5
I
6
Figure E 8: Projected future position of continuous incident step function waveform in absence of boundary at x = 0.
2
Ci
units
time step
units
time step
C2
xii(4)
4 units
xi 2(4)
6 units
Xr1(4) = -Xii(4)
-4 units
Xr2(4) = -Xi2(4)
-6 units
xti(4) = -'X1i(4)
2 units
Xt2(4) = -Xi2(4)
3 units
Cl
C,
Rg = Tk
1
L
n n
2
u
Table E 2: Required
values to solve continuous step function example.
Ta eE_:_rdestos
121
Amplitude
t =4
- 1
IcI=1
c =2
-0.5
I
I
I
II
-3
-2
-1
I
-6
-5
-4
II
1
2
3
4
5
6
Figure E 9: Resultant waveforms (continuous step function example).
Again, the correctness of these waveform locations is veri fied by showing each time step of the reflection
and transmission processes, individually.
122
t=0
Amplitude
c=2
-1
~
I
~
-6l
-I
-5
-6
-4
I----
I
-3
-2
X
-1F
-1
1
2
3
4
6
5
t= 1
Amplitude
c,=2
-1
-3
-4
-5
-6
-2
1
-1
2
3
4
6
5
Ampli tude
t=2
-1
=2
-0.5
-
-II
-5
-6
I
-3
I
-4
-1
-1
-2
1
2
3
4
6
5
t= 3
Amplitude
-1
c =2
-0.5
--
-II
-5)
-6
I
-4
I
-3
2
-2
1
-1
-1
S 2x
14560
Amplitude
t=4
A
- 1
c =2
I
~-i
I
-6
-0.5
I
I
I
-5
I
I
-4
I
I<
I
-3
I
I
I
I
-2
-1
Tj
2
Figure E 10: Continuous stepfunction waveform reflection in individual time steps.
123
3
4
5
6
Amplitude
t1
c =2
<!
t=Q
I
I
1
-2
-3
-4
-5
-6
I
I
I
-I
I
1
4
3
3
2
1
-1
I
5
5
6>
6
t =1
Amplitude
c =2
- 1
~
-6
I~
I
I
I
I
-5
-4
-3
-2
-1
4
3
2
1
6
5
Amplitude
t=2
1C>=1
0.5- II
-5
-
-
I
II
I
1
-
II
-3
II
I
-4
I
~~-I
-.-. I
-6
-1
-2
I
1
2
2
I
3
I
I
I
I,.~
4
5
I
6
t =3
Amplitude
A
-1
c =1
-0.5
.,-I
- 6
-6
I
I
-5
I
I
-4
I
I
-3
I
I
I
-2
-1
-1
3
2
4
Amplitude
6
5
t=4
A
- 1
c =1
6
-0.5
sJ~~L
I
gI
-6
-5
I
I
-4
I
I
-3
I
I
I
I
-2
-1
I
12
3
Figure E 11: Continuous step function waveform transmission in individual time steps.
124
4
5
6>
x
A final example, a continuous sinusoidal incident waveform, is illustrated in the exact same manner. Again,
for this example, let cl = 2,
c2 =
1, and R = Tf = 1. Here, five x-coordinates are specified to define the
location and shape of the sinusoidal waveform, given by the peaks and x-intercepts of the incident
waveform. All of the values required to evaluate Eqs E14 and E15, and the corresponding results, are
summarized in Table E3.
Amplitude
t=o
c,=2
-- 1
-6
|
|
-5
-4
'I
-3
I
ii
-2j-1
1
i
2
3
I
4
5
6
_+ -1
IV
Figure E 12: Continuous incident sinusoidal waveform.
t=3
Amplitude
A
c,=2
- 1
I
-6
-5
I
I
I
I
I
I
I
I
-4
-3
-2
-1
1
2
3
4
6
-- 1
Figure E 13: Projected future position of continuous incident sinusoidal waveform in the absence of a boundary at x = 0.
125
units
2
c1
time step
1 units
C2
time step
3 units
3.5 units
xii(3)
xi2 (3)
4 units
Xi3(3)
4.5 units
5 units
-3 units
-3.5 units
X14(3)
xi 5(3)
Xr1(3)
=
-xii(3)
Xr2(3) = -Xi2(3)
=
Xr3(3)
-4 units
-xi 3 (3)
-4.5 units
Xr4(3) = -i4(3)
= -xi 5(3)
-5 units
xti(3) = Exjj(3)
1.5 units
Xt2(3) = -Xi2(3)
1.75 units
Xrs(3)
C-1
C1
xt 3 (3) =
2 units
x1 3 (3)
C1
2.25 units
Xt4(3) = CXi4(3)
C1
xts(3) =
2.5 units
-x's(3)
C1
RT = T
Table E 3: Required values to solve contin uous sinusoidal waveform example.
Amplitude
t=3
c2 =1
C =2
-
-
K^-
I
-6
t0.
-5
4
-5
I
I
I
-2
I
-1
5
I
(')
-- 0.5 1
2
I]
3
I I
I
I-,
4
5
6
Figure E 14: Resultant waveforms (sinusoidal waveform example).
The resultant waveform locations are verified with the individual time steps of the reflection and
transmission processes, as in the previous two examples.
II
126
t=o
Amplitude
A
c,=2
-1
I
-. I
-6
I
-5
1
-3
-4
1
1
-2U
2
3
4
6
5
-- 1
t=1
Amplitude
0.5
x
-6
-5
-4
-3
-2
-1
--
0.5 1
2
3
4
6
5
t=2
Amplitude
c =2
-6
-5
I
I
-4
-3
-0.5
K
2
~7-0.5 1
-1
2
3
4
t=3
Amplitude
c =2
K>0"
-5k
4
-J
I
I
-2
-1
L
6
5
0.5
t-0.5
S
2
Figure E 15: Continuous reflected sinusoidal waveform in individual time steps.
127
3
4
A
b
t=
Amplitude
c =2
- 1
I
-6
-5
-4
1
-3
2
3
4
>
x
>
x
6
5
---1
t=1
Amplitude
c,=2
- 1
-6
_I-5I
I
-4
I
-3
1
-2
11
-1 -0.5-
I.
-1-4-
t=2
Amplitude
c1
0.5-
I
-5I
I
-4I
I
%-
-I
I
II
-I
II
I
-6
-5
-4
-3
-2
-1-0.5-
I
1U
2
3
4
6
5
NV
t=3
Amplitude
A
-0.5
-6
I
-5
I
-4
II
-3
II
-2
II
I--
0.5
25
-1
V
Figure E 16: Continuous transmitted sinusoidal waveform in individual time steps.
128
Appendix F: The Importance of the Coupling
The selection of a coupling agent between the transducer/ receiver and the test specimen is
crucial to the NDE process. Extensive research into optimization of various coupling techniques, materials,
and configurations continues even today, and a basic library search of keywords like "ultrasonic testing"
and "coupling" will yield results about investigations of inductive and capacitive coupling systems, the
effects of coupling variation on NDE processes, and various non-contact coupling schemes, among other
fascinating topics. Indeed, a full discussion of the importance of the specimen-transducer coupling is a
topic worthy of a thesis in its own right.
One important aspect of the selection of an appropriate coupling is that it allows for calculated
manipulation of the pulse that impinges upon the test piece. If the designers of an ultrasonic test
particularly concern themselves with the material properties of the coupling as compared to that of the
test piece, the characteristics of the wave pulse can be strategically manipulated to yield a desired result.
Since the speed of a propagating wave in a given material is based solely upon the material properties of
the particular propagating medium (ignoring the effects of wave dispersion), the relative wave speeds of
two different materials in a wave transmission will affect the overall shape of the transmitted wave, as
demonstrated below. Supposing that the two materials involved in the wave transmission are the coupling
agent and the test piece itself, the same changes in the pulse shape occur, and the designer of an
ultrasonic NDE test setup can choose a coupling agent with specific changes to the pulse shape in mind.
Spatial Dilation and Contraction
One such manipulation is the spatial dilation or contraction, or relative lengthening or shortening,
of the wave pulse. Consider the wave pulse shown in Figure F1, incident upon a coupling/ test piece
junction at x = 0.
129
Test Piece
Coupling Material
t=to
x=O
x
=
x
Figure F 1: Wave pulse incident upon coupling/ test piece junction.
At time t = ti, the wave pulse reaches the junction and, when this contact occurs, some of the
pulse's energy is transmitted through the junction into the second medium while the remainder is
reflected back towards the pulse's point of origination. However, the mismatch between the wave speeds
in the two media results in the portion of the pulse that has already been transmitted into the second
medium moving either faster or slower than the portion that has yet to be transmitted and still remains
in the original medium of propagation. In the event that the wave speed in the second medium is faster
than that of the first, the overall affect is a relative stretching or lengthening of the wave pulse, as the
transmitted portion "outruns" the remainder of the pulse yet to be transmitted through the boundary.
However, if the wave speed in the second medium is slower than that of the first, the opposite affect is
observed and the transmitted wave pulse is spatially compressed when compared to the original (incident)
pulse.
130
C test piece > C coupling
Test Piece
Coupling Material
t=to
x
x =0
Test Piece
Coupling Material
x
x=O
Test Piece
Coupling Material
t =t 2
x
x =0
Test Piece
Coupling Material
t=t
x
x
L0
Figure F 2: Coupling-to-test piece transmission with spatial dilation.
131
3
C test piece -> C coupling
Test Piece
Coupling Material
t=to
x
x=0
Test Piece
Coupling Material
t=tl
x
x=0
Test Piece
Coupling Material
t=t 2
x
x =0
Test Piece
Coupling Material
t =t3
x
x =0
Figure F 3: Coupling-to-test piece transmission with spatial contraction.
132
Mathematically, this spatial dilation/ contraction is found in the generalization of the argument
of the wave function defining the transmitted wave. The boundary condition of continuity of displacement
at the junction dictates that:
Gt(0, c 2 t) = ft(-c 2 t) = Tgfi (-c 2 t)
where ft and
Eq (Fl)
f' are the generalized wave functions of the transmitted and incident wave pulses,
respectively, c2 and cl are their respective wave speeds, and T is the wave pulse transmission coefficient.
When the arguments of the wave functions in Eq Fl are generalized and expanded to include all values of
x:
ft (xt, c2 t)
Thus we see that for c1
>
=
Tfi
(2Xi,
c2 t)
Eq (F2)
C2 the wave pulse is spatially contracted, while for c1< c2 it is spatially dilated.
In practice, this phenomena can be used to improve an ultrasonic test and collect data more
efficiently. For example, for a generated pulse that is spatially narrow, the coupling agent can be selected
such that the pulse transmitted into the test piece is elongated, allowing for more subtle details of the
reflected pulse to be readily observed. This might be useful in industry, where it might be simpler to
change the coupling agent than it would be to vary the characteristics of the generated pulse.
Mechanical Attenuation and Amplification
The other pulse characteristic that is modified when a pulse traverses a boundary between two
dissimilar materials is the amplitude of displacements. This is readily seen, mathematically, in the relation
given above in Eq F2. Namely, the transmission coefficient, T , either amplifies or reduces the relative
amplitude of the transmitted wave pulse with respect to the incident pulse. In the event that T > 1,
amplification occurs. Otherwise, the pulse is attenuated as the pulse amplitude is reduced by a factor of
T . To see how the selection of an appropriate coupling agent can achieve these results directly, the
definition of the transmission coefficient is analyzed.
=
2E AIc
E1Alc 12 +E 22A 2c1
Eq (F3)
For simplicity, assume A, = A2 = A for the coupling - test piece boundary so that all of the cross-sectional
area terms cancel. This is a valid assumption since it is absolutely feasible to fabricate this condition in
practice. Thus, only the Young's modulus and wave speed terms remain and, with some basic algebraic
manipulation, it is easy to find the conditions necessary for wave pulse amplification to occur.
133
2E 1 AIc 2
E1 A 1IC2 +E 2 A 2 C1
>
2Elc 2 > Elc2 + E 2 c1
Elc 2 > E 2c1
E,
E 2 El
L>
P2
P1
E 1 E2 > EE2
Pi
P2
P>
1
Eq (F4)
P2
Similarly, satisfying the opposite condition (L> 1
Pi
would yield signal attenuation. Thus, the desired
result can be obtained directly simply by selecting the coupling agent based upon its material density
relative to that of the test piece.
It is also fairly straightforward to see why these conditions would yield signal attenuation or
amplification when the actual mechanism of wave propagation is considered. Since the wave propagates
by causing oscillations in adjacent elemental masses of the propagating medium, it makes sense that in a
denser medium it would be more difficult to move the particles and cause oscillations. Thus, if the second
medium is denser than the first, particle oscillations will be smaller in the second medium than in the first
with the same amount of wave energy17 and the signal is attenuated. Similarly, if the first medium is
denser than the second, the particles of the second medium will move more freely causing larger
oscillations after wave transmission, and the signal is amplified.
Just as in the case of spatial dilation or contraction, signal amplification or attenuation achieved
in this way might be desirable if the generated signal is either too weak or the amplitude is too large to
use for ultrasonic testing, and it is either too costly or overly complicated to vary the pulse generation
directly. In this case, mechanical amplification or attenuation through the selection of an appropriate
coupling agent is another alternative that might prove useful and cost efficient.
11 While the coupling - test piece boundary will cause both reflection and transmission, as before, couplings are
specifically chosen so that the majority of the wave is transmitted and very little is reflected back into the coupling.
Thus, most of the wave's energy will be transmitted into the test piece and the energy is approximately the same
between the incident wave and the transmitted wave.
134
Bibliography
.
[1] D. E. Bray and R. K. Stanley, Nondestructive evaluation : a tool in design, manufacturing, and service
/ Don E. Bray, Roderic K. Stanley. Boca Raton : CRC Press, c1997., 1997.
[2] L. Cartz, Nondestructive testing : radiography, ultrasonics, liquid penetrant, magnetic particle, eddy
current/by Louis Cartz. Materials Park, OH : ASM International, c1995., 1995.
[3] S. Sokolov, "US Patent 2,164,165 Means for Indicating Flaws in Materials," 27-Jun-1939.
[4] "Ultrasonic testing," Wikipedia, the free encyclopedia. 03-Dec-2014.
[5] F. A. Firestone, "Flaw detecting device and measuring instrument," US2280226 A, 21-Apr-1942.
[6] A. P. French, Vibrations and waves. New York: Norton, 1971.
[7] "History of Ultrasonics." Introduction to Ultrasonic Testing. N.p., n.d. Web. 03 Dec. 2014.
[8] James H. Williams, Jr., "Wave Propagation: An Introduction to Linear Analyses," Massachusetts
Institute of Technology, Cambridge, MA.
135