DIFFRACTION EFFECTS ON ULTRASONIC GUIDED WAVES
RADIATED OR RECEIVED BY TRANSDUCERS MOUNTED ON
THE SECTION OF THE GUIDE
K. Jezzine and A. Lhémery
Commissariat à l’Énergie Atomique, LIST, CEA-Saclay, bât. 611, 91191 Gif-sur-Yvette
cedex, France
ABSTRACT. The Semi-Analytic Finite Element method (SAFE) has been used to model the
propagation of ultrasonic waves guided (GW) by a structure and radiated at the guiding surface, a
classical configuration for nondestructive testing (NDT) of large structures. Here, the SAFE method is
derived to deal with configurations where ultrasonic source / receiver are mounted on the guide
section. It allows to predict very efficiently amplitudes and waveforms of propagating,
inhomogeneous and evanescent modes in such configurations. The model is used to study transducer
diffraction effects and their implications to GW-NDT.
Keywords: guided waves, UT simulation, mode selection, semi-analytic finite elements
PACS: 43.20.Gp – 43.20.Mv – 43.35.Yb – 43.35.Zc
INTRODUCTION
Ultrasonic guided waves (GW) propagate over long distances in guiding structures
such as plates, rods, tubes etc. Therefore, they can be used for the inspection of large
structures in testing configurations where neither the ultrasonic source nor the receiver must
be moved along the guiding structure to image the whole structure [1-2]. However, two
characteristics of guided elastic wave propagation make difficult the interpretation of
experimental data: first, at a given excitation frequency, several modes can co-exist in the
structure and second, these modes propagate at different speeds and these speeds are
frequency-dependent (dispersion). Various models to predict them in geometrically
canonical waveguides are well established [2-4]. Dispersion curves can be calculated
depicting the behavior of propagating modes as functions of the frequency [4-5].
Predicting the amplitude of all propagating modes as they are radiated by an actual
transducer is somewhat more difficult. Thanks to progresses during the last decade in the
development of simulation tools able to make such predictions, it is now easier to interpret
actual measurements and even better, to design testing configurations in which known
specific modes are radiated in the structure [4].
GW are commonly radiated from the guiding surface: one of the two planes of a
plate, cylindrical surface of a rod, outer (or inner) cylindrical surface of a tube etc. Among
various modeling approaches, the semi-analytic finite element (SAFE) method allows such
predictions [6-8].
It is sometimes more convenient to radiate (and receive) GW from the guide section
transducer
waveguide
guide section
S
z
x is
guide a
FIGURE 1. Configuration in hands: a transducer is mounted on the (arbitrary) section of a waveguide.
(as shown on Fig. 1) depending on the accessibility of the structure to an ultrasonic
transducer (e.g., a guiding structure embedded in a solid). To deal with such testing
configurations, a modified form of the SAFE method is proposed in the present paper. It
allows to predict very efficiently (the 3D problem being transformed into a 2D one)
amplitudes and time-dependent waveforms (by Fourier synthesis) of propagating, as well as
both inhomogeneous and evanescent modes.
The paper is organized as follows. First, we review existing methods to predict the
ultrasonic field radiated in a waveguide in configurations where transducers are mounted on
the guide section. Then, an adaptation of the SAFE method is proposed to deal with
arbitrary guide section in such a configuration. A validation of the theory is then made by
comparison with theoretical results of the literature. Finally, the model is used to simulate
transducer diffraction effects (e.g. influence of transducer bandwidth and aperture).
Numerical examples given concern modal radiation and reception in cylindrical guides.
THEORY
Since in most common GW-NDT configurations, radiation and reception of GW are
made from or through the guiding surface of the structure, most of the literature where the
modeling of source and receiver behaviors has been addressed is limited to these cases.
Configurations of guided wave radiation or reception by transducers mounted on the
section of the guiding structure pose specific problems.
Review of the Literature
The ultrasonic field in a waveguide can be expressed as an infinite sum of the
existing modes at a given frequency, as
 v ( x, y, z )   An v n ( x, y, z )   An ~
v n ( x , y ) e j ( k n z  t )

n
n
,
(1)

j ( k n z  t )
~
σ ( x, y, z )   Anσ n ( x, y, z )   An σn ( x, y )e
n
n

~ and An are the
where v is the displacement and σ the stress at coordinates (x,y,z), kn, ~
vn , σ
n
th
wave number, displacement, stress and amplitude associated with the n mode.
The end conditions imposed by a transducer mounted on the guide section (assumed
to be perpendicular to the guide axis) can be either mixed or pure. For mixed end
conditions, either tangential displacement and normal stress , or, normal displacement and
tangential stress at z=0 are prescribed whereas for pure end conditions, either both
components of displacement, or, both components of stress at z=0 are prescribed. 
~ and kn is
Deducing the coefficient An for a given mode knowing ~
vn , σ
n
straightforward in the case of axisymmetry and mixed end conditions [9-10]: the use of
Fraser’s orthogonality relation [11] leads to a simple expression for An written as
a
~
Qmn   (v~r ( m)~rz ( n ) v~z ( n )~zz ( m) )rdr  0 except if n  m ,
(2)
0
a
1
An  ~  (vr ( source) σ~rz( n ) v~z ( n ) zz( source) )rdr .
Qnn 0
(3)
Nevertheless, the physical behavior of a piston-like piezoelectric transducer directly
coupled to an elastic medium is accurately modeled as a pure end condition of imposed
normal stress. In this case, one needs to minimize the residual boundary values, i.e. the
difference between the prescribed tractions and the modal series [12-13] (the series being
infinite, it is truncated in practice to N terms). This is expressed by
N
N
n 1
n 1
0   An σ~rz(n), σ~zz(source)   An σ~zz(n).
(4)
The resolution of (4) requires the knowledge of the modal decomposition. For
canonical geometries (plates, rods or tubes), the dispersion equation that relates the wave
number and frequency is known analytically. The wave numbers are then computed using a
root-finding routine. This approach may be implemented for multi-layered structures [5].
One drawback to this method is that it can not deal with guides of arbitrary section.
Dispersion in complex structures has been studied by various authors using the SemiAnalytical Finite Element method (SAFE) [14-17]. In this method, the section of the guide
is meshed by finite elements whereas wave propagation in the axial direction is treated
analytically. Multilayered structures, anisotropy, viscoelasticity can be easily accounted for.
SAFE has been used for configurations where the excitation modeled as an external
force is made from the guiding surface [18-20]. To our knowledge, excitation by a
transducer mounted on the guide section has not been studied using SAFE calculation.
Nevertheless, this problem is closely related to that of the scattering from the free end of a
semi-infinite guide which has been studied by several authors [21-23]. Indeed, the
excitation in this case may be modeled as a vertical boundary condition [24] as shown later.
The Semi-Analytic Finite Element Method (SAFE)
The model derived in our study applies to waveguides of arbitrary section and
anisotropy. However, to simplify the writing of equations and since all the examples given
thereafter concern cylindrical guides made of isotropic materials, axial symmetry and
isotropy will be assumed below. This implies that solutions are also of axial symmetry.
The section is meshed by annular (1D computations considering the symmetry)
three node elements, as shown on Fig. 2. The displacement field on the ith element writes
~(r , z, k ;[i])  N(r ) d i e j ( kz t ) ,
u
(5)
where N (r ) is a matrix of quadratic interpolation functions, di is the unknown vector of
nodal displacements and k is the unknown wavenumber. Using this formalism, the
application of the virtual works principle leads to the following quadratic eigenvalue
system for the ith element:
(K 1i  jkK i2  k 2K i3 ) d i   2M i di  0 ,
(6)
where the various K are matrices of rigidity and M the mass matrix. Since nothing original
is presented in the present paper about general SAFE theory, we do not develop the detailed
r
r
r
En
En-1
E1
z
FIGURE 2. Meshing of the circular section by means of annular elements {1,…, n}. White (resp. black) dots
denote element edge nodes (resp. center). Thanks to the symmetry, the 2D computation becomes a 1D one.
formulation that can be found in Ref. [7].
After assembling every element systems and solving the resulting quadratic system
of 2M equations, one obtains 4M eigenvalues and eigenvectors (M being the number of
nodes) which correspond to the modal solution of the structure.
Case of a Source Mounted on the Guide Section
As the source is mounted on the guide section at z = 0, one selects among the 4M
computed by Eq. (6), those physically admissible as far as only positive values of z are
concerned. For propagating modes (purely real k), one keeps modes that transmit energy in
the +z direction. For evanescent or inhomogenous modes (pure imaginary or complex k),
eigenvalues with positive imaginary part are kept. In the end, 2M modes are selected.
The stress is deduced from the eigenvector displacement at the M nodes. The emitter
behavior is accounted for by explicitly introducing terms of stress sources. For this, the sum
of stresses of the modes radiated in the structure must be equal to the prescribed (source)
stresses. A similar approach has been used to solve the problem of reflection at the free end
of a semi-infinite guide [21-23] where the incident and reflected mode series must match
the free-end boundary condition. The series of mode amplitude An is obtained by solving
the 2M x 2M linear system where the stress series are expressed at nodes [24]:
2M
σ zz(source)(ri , z  0)   An σ~zz(n)(ri ),
n 1
2M
0   An σ~rz(n)(ri ) , for i = 1,…, M.
(7)
n 1
Actual transducer excitation signals are defined over a finite bandwidth that may
vary from one NDT application to another. Prediction of transducer diffraction effects in
the time domain must therefore take into account a certain frequency spectrum. This is
simply done by Fourier synthesis. Note that the overall system [Eqs. (6-7)] is frequencydependent but all the matrices are dependent only on guide geometry and elastic properties
so that they can be computed once and then re-used when computing modes at the various
frequencies of the excitation spectrum.
RESULTS
Comparison with Exact Results in a Canonical Configuration
The dispersion relation for axially symmetric waves in a stress free cylindrical rod
TABLE 1. Wavenumbers (in m-1) computed by present model for various numbers of elements of the 9
propagative modes in a steel cylinder 20-mm-Ø at 1 MHz, compared to roots of Pochhammer’s solution.
mode
5 elts.
10 elts.
20 elts.
30 elts.
exact
L(0,1)
2058.4
2094.1
2101.3
2101.9
2102.0
L(0,2)
1883.1
1884.1
1884.2
1884.3
1884.3
L(0,3)
1753.1
1757.3
1757.6
1757.6
1757.6
L(0,4)
1525.9
1548.9
1550.5
1550.5
1550.5
L(0,5)
1174.2
1247.6
1253.4
1253.6
1253.6
L(0,6)
1035.8
1047.7
1049.4
1049.4
1049.4
L(0,7)
951.4
988.1
991.3
991.5
991.4
L(0,8)
784.1
866.1
870.2
870.4
870.4
L(0,9)
557.1
619.8
623.4
623.5
623.4
is known analytically as the Pochhammer-Chree equation. It is possible to compute its roots
(wavenumbers) numerically at a given frequency. Table 1 compares values obtained from
Pochhammer’s equation to those obtained with the present model considering a frequency
of 1 MHz at which nine real propagative modes co-exist in a 20-mm-Ø cylinder made of
steel. Clearly, the use of only five elements leads to large inaccuracies (up to 10% for one
of the modes). The use of 10 (resp. 20, 30) elements leads to predicted values with an error
less than 0.6% (resp. 0.3‰, 0.16‰).
In a very recent paper by Puckett and Peterson [13], comparison of waveforms
experimentally measured and theoretically predicted was given. To test our own approach,
we have reproduce some of their results. Figure 3 compares one of their experimental
results to our prediction. The configuration is shown on the top of the figure: two 25-mm-Ø
transducers are mounted on the opposite sections of a 250-mm-long cylinder made of fused
quartz of the same diameter. One works in the transmit-mode whereas the second works as
a receiver. A narrowband (6.7% of relative bandwidth) excitation pulse of 1107 kHz center
frequency is considered. At this center frequency, 12 propagative modes co-exist.
Since the configuration involves a reception process, field results computed using
the present model must be post-processed. We consider, using reciprocity arguments the
receiver as uniformly sensitive to the normal component of the particle velocity on the
section, this elastodynamic quantity being simply integrated over the receiver surface as
a
O(t )   v z (r , t ) r dr .
(8)
0
Comparison is very satisfactory and this constitutes an element of validation of our
approach to quantitatively predict transducer diffraction effects on mode amplitudes.
L
Ø
d
transducer E
40
60
80
100
cylindrical guide
120
measured (Puckett and Peterson)
140
40
transducer R
60
80
100
120
140
simulated (present model)
FIGURE 3. Top: configuration considered in [13]. Bottom: left, measured signal in [13]; right: prediction
using present approach. The diameter of the transducer working in the transmit mode equals that of the guide.
a)
b)
c)
FIGURE 4. Same configuration as in Fig. 3. The transducer is excited by three different Guassian pulses
having the same center frequency of 1107kHz but different relative bandwidth. a) 6.7%, b) 20%, c) 40%.
Effects of Transducer Bandwidth on Radiation
We show now results (Fig. 4) for the same configuration where the bandwidth of the
excitation pulse is varied. Two pulses of the same center frequency of 1107 kHz are
considered, one having a relative bandwidth of 20%, the other of 40%.
Effects of Transducer Aperture on Radiation
The effect on radiation of different source diameters (25-mm-Ø, 12.5-mm-Ø and
point-like excitation) on the wave field is now considered in Figs. 5 and 6. In Fig. 5, the
time-dependent waveforms The various waveforms taken at points along a radius of the
guide at a distance of 250mm from the sources are displayed as pseudo- B-scan images.
field
points
z
d
960 µs
Ø
point-source
uz (r, z = 250mm, t)
Ø = 12.5 mm
400 µs
Ø = 25 mm
200 µs
0
mm
12.5 0
mm
12.5 0
mm
12.5
FIGURE 5. Time-dependent normal displacement fields radiated by a 25-mm-Ø transducer (left), 12.5-mmØ one (middle), point-like one (right).
a)
b)
c)
40
60
80
100
120
140
160 µs
FIGURE 6. Three different transducers of 25-mm-Ø (a), 12.5-mm-Ø (b), point-like (c) are used as
transmitters excited by a narrowband pulse of 1107kHz center frequency and 6.7% relative bandwidth radiate
fields in the25-mm-Ø fused-quartz guide measured by a of receiver (same Ø) at a distance of 250mm.
L(0,1)
L(0,2)
L(0,3)
L(0,4)
1
0
0
1
0
1
0
uz(r)
1
a
ur(r)
-2.5
0
0
-4.5
0
a
-3
0
1.5
a
-6
0
1
0
0
-3
-2.5
0
a
0
a
a
-3
0
1.2
0
-0.2
0
a
a
FIGURE 7. Profiles at z=0 of modes radiated in a cylindrical waveguide made of steel of 20-mm-Ø at a
frequency of 500kHz. Black lines: traction free case. Gray lines: embedded in cement. Top (resp. bottom):
maximum amplitude of axial (resp. radial) displacement.
SAFE Calculations in an Embedded Waveguide
Guided waves in embedded waveguides have many industrial applications, one of
them being the inspection of steel tendons used for the post-tensioning in concrete
structures. Embedded waveguides have been studied by various authors (see for exemple
Ref [25]). The simulation must account for leakage of the previously free modes in the
surrounding medium. For this purpose, we propose the
TABLE 2. Wavenumbers (in m-1)computed by present model of the four propagative modes in a steel
cylinder either free (top) or embedded in ciment (bottom) of diameter 20 mm at a frequency of 0.5 MHz.
modes
free
embedded
L(0,1)
1049.8
1045.6 – 34.6 i
L(0,2)
818.9
819.0 – 8.5 i
L(0,3)
562.1
561.7 – 14.6 i
L(0,4)
471.4
471.7 – 6.2 i
SUMMARY AND FUTURE WORK
A model has been proposed, based on the Semi-Analytic Finite Element method to
compute ultrasonic guided waves radiated (and by reciprocity received) by a transducer
mounted on the guide section. Its most important feature is that it only requires
computations in the section of the guide while propagation in the direction of the guide axis
is handled by propagators of analytic nature. Therefore, the ultrasonic field radiated at any
position in the 3D guide can be predicted by efficient 2D computations (1D in case of axial
symmetry). Sections of arbitrary geometry can be considered. The ultrasonic transducer
mounted on the guide section is modeled as a source of normal stress (pure conditions), a
common assumption for transducers directly coupled to a structure. Using the model;
typical transducer diffraction effects have been studied: mode selection depending on
excitation bandwidth, influence of transducer aperture on the relative amplitude of coexisting modes etc. This study emphasizes the role of transducer characteristics in
quantitative NDE methods based on guided waves applied to the inspection of large
structures and the importance of the use of appropriate simulation tools to accurately
simulate their effects.
REFERENCES
1. Cawley, P., Review of Progress in QNDE, Vol. 22, eds. D. O. Thompson and D. E.
Chimenti, AIP Conference Proceedings 657, Melville, 2003, p. 22.
2. Rose, J. L., Ultrasonic waves in solid media, Cambridge University Press, Cambridge,
U.K., 1999, Chap. 8-16.
3. Zemanek, J., J. Acoust. Soc. Am. 51, 265 (1972).
4. see for instance papers published in this series by (among others) Pr. P. Cawley’s group
at Imperial College, London, U.K. and Pr. J. L. Rose’s group at Penn State, USA.
5. Lowe, M. J. S., IEEE Trans. Ultrason. Ferroelectr. Freq. Control 42, 525 (1995).
6. Liu, G. R. and Achenbach, J. D., J. Appl. Mech. 62, 607 (1995).
7. Hayashi, T., Song, W. J. and Rose, J. L., Ultrasonics 41, 175 (2003).
8. Galán, J. M. and Abascal, R., Int. J. Numer. Meth. Engng. 58, 1091 (2003).
9. Duncan Fama, M. E., Quart. J. Mech. Appl. Math. 25, 479 (1972).
10. Herczynski, A. and Folk, R. T., Quart. J. Mech. Appl. Math. 42, 523 (1989).
11. Fraser, W. B., J. Sound Vib. 43, 568 (1975).
12. Gregory, R. D. and Gladwell, I., Quart. J. Mech. Appl. Math. 42, 327 (1989).
13. Puckett, A. D. and Peterson, M. L., Ultrasonics 43, 197 (2005).
14. Dong, S. B. and Nelson, R. B., J. Appl. Mech. 39, 739 (1972).
15. Gavrić, L., J. Sound. Vib. 185, 531 (1995).
16. Hayashi, T. and Rose, J. L., Mat. Eval. 61, 75 (2003).
17. Damljanović, V. and Weaver, R. L., J. Acoust. Soc. Am. 115, 1572 (2004).
18. Liu, G. R. and Achenbach, J. D., J. Appl. Mech. 62, 607 (1995).
19. Zhuang, W., Shah, A. H. and Dong, S. B., J. Appl. Mech. 66, 665 (1999).
20. Hayashi, T., Kawashima, K., Sun, Z. and Rose, J. L., J. Acoust. Soc. Am. 113, 1241
(2003).
21. Rattanawangcharoen, N., Shah, A. H. and Datta, S. K., J. Appl. Mech. 61, 323 (1994).
22. Taweel, H., Dong, S. B. and Kazic M., Int. J. Sol. Struct. 37, 1701 (2000).
23. Galán, J. M. and Abascal, R., Int. J. Numer. Meth. Engng. 53, 1145 (2002).
24. Liu, G. R. and Achenbach, J. D., 61, 270 (1994).
25. Beard, M. D., Guided wave inspection of embedded cylindrical structures, PhD thesis,
Imperial College, London (2002).
field
points
z
d
960 µs
Ø
point-source
uz (r, z = 250mm, t)
Ø = 12.5 mm
400 µs
Ø = 25 mm
200 µs
mm
0
mm
12.5 0
field
points
z
d
12.5 0
mm
12.5
960 µs
Ø
point-source
ur (r, z = 250mm, t)
Ø = 12.5 mm
400 µs
Ø = 25 mm
200 µs
0
mm
12.5 0
mm
12.5 0
mm
12.5
FIGURE 5. Time-dependent normal (top) and radial (bottom) displacement fields radiated by a 25-mm-Ø
transducer (left), 12.5-mm-Ø one (middle), point-like one (right). The various waveforms taken at points
along a radius of the guide at a distance of 250mm from the sources are displayed as pseudo- B-scan images.
L
Ø
d
transducer E
40
60
80
100
cylindrical guide
120
measured (Puckett and Peterson)
140
40
60
transducer R
80
100
120
simulated (present model)
FIGURE 3. Top: configuration considered by Pluckett and Peterson [13]. Bottom: left, measured signal by
Puckett and Peterson; right: prediction using the present approach. In this example, the diameter of the
transducer working in the transmit mode equals that of the guide.
140
L(0,1)
1
L(0,2)
1
0
L(0,4)
1
0
uz(r)
0
L(0,3)
1
0
a
ur(r)
-2.5
0
0
-4.5
0
a
-3
0
1.5
a
-6
0
1
0
0
-3
-2.5
0
a
0
a
a
-3
0
1.2
0
-0.2
0
FIGURE 7. Profiles at z=0 of modes radiated in a cylindrical waveguide made of steel of 20-mm-Ø at a
frequency of 500kHz. Black lines: traction free case. Gray lines: embedded in cement. Top (resp. bottom):
maximum amplitude of axial (resp. radial) displacement.
a
a
a)
b)
c)
40
60
80
100
120
140
160 µs
FIGURE 7. Three different transducers of 25-mm-Ø (a), 12.5-mm-Ø (b), point-like (c) are used as
transmitters excited by a narrowband pulse of 1107kHz center frequency and 6.7% relative bandwidth radiate
fields in the25-mm-Ø fused-quartz guide measured by a of receiver (same Ø) at a distance of 250mm.
a)
b)
c)
FIGURE 7. Same configuration as in Fig. 3. The transducer is excited by three different Guassian pulses
having the same center frequency of 1107kHz but different relative bandwidth. a) 6.7%, b) 20%, c) 40%.
0
amplitude in dB
-10
-20
-30
-40
-50
-60
0
0.2
0.4
0.6
0.8
1
z/Ø
FIGURE 7. Variation with z in the very near-field of the maximum amplitude of the axial displacement at
r=0 of the different modes radiated by the transducer (same Ø as that of fused-quartz waveguide). Solid (resp.
dotted, dashed) lines: propagative modes (resp. inhomogeneous, evanescent).
r
visco-elastic absorbing layer
cement
steel
FIGURE 7. In the embedded case, the meshing os applied to the guide (steel), the cement in which the guide
is embedded and the visco-elastic absorbing layer used to absorb the energy leaked in the cement.