Viewer-controlled computer images for some
basic wave phenomena
R. Dean Ayers and Nader Inan
Department of Physics and Astronomy, California State University, Long Beach
1250 Bellflower Boulevard, Long Beach, California 90840-3901
rdayers@csulb.edu
Abstract: Textbooks of the future are likely to be electronic and to
include many features not available in printed books. Some of the most
striking changes might be found in the figures used to support the text
and equations. The teaching of acoustics could benefit considerably from
those changes. A few examples of computer graphics that can be used to
introduce basic concepts about wave propagation are presented here.
Implications for possible changes in the content and organization of
courses that use these graphical tools are also discussed.
©2002 Acoustical Society of America
PACS numbers: 43.10.Sv, 43.20.-f, 43.75.-z
Date Received:
Date Accepted:
1. Introduction
Graphical representations of a physical system and the relevant mathematics can be very
helpful in defining a problem, guiding the mathematical development, and seeing that the
results make sense. Sketches on paper serve as reminders and checks for the mental
images that they represent. Software for generating and controlling images on the screen
of a personal computer provides the additional benefits of greater accuracy and precision.
Figures can be set in motion, both for including time as an independent variable and for a
systematic variation of parameters. Three-dimensional shapes may also be examined from
many orientations. Providing the viewer with controls to manipulate live images on the
screen makes the presentation more concrete and encourages active mental involvement.
There are several types of software that can be used to generate the figures
discussed in this article. We have used DPGraph because it has been designed for speed
and for easy sharing of files from a Web site. In order to display and manipulate the
figures for this article, you will need to download the free DPGraph Viewer by the methods
described in the appendix. General instructions on the use of the scrollbar and other
controls are also found there. Working from a printout of this text is convenient for
following the steps to manipulate and interpret each figure while it is displayed on your
computer. You could also reduce the windows for the text and the figure in order to
display both simultaneously.
2. “Lurching waves”
Introductory treatments of waves tend to focus on undamped traveling sinusoidal waves
and ideal standing waves with perfect nodes. These are just special cases from the
continuum of more interesting and realistic wave behaviors that we will examine here.
Open Mm. 1, and refer to the appendix for use of the controls mentioned below.
At first we see the second normal mode of vibration for a simple, ideal structure. The
dependent variable y might be transverse displacement on a string that is tied to two rigid
supports, or perhaps the longitudinal displacement of air in a cylindrical pipe with both
ends closed. Selecting variable A under the scrollbar menu and moving the slider upward
by “jumps” displays the following sequence of plots: A = 1, a rightward traveling wave,
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y(x,t) = cos(kx-t); A = 2, a leftward traveling wave, y(x,t) = -cos(-kx-t); A = 3, both of
those; A = 4, both traveling waves together with their sum, which was shown by itself at
first (A = 0). To speed up or slow down the motion, select variable D and change it with
the slider. You may want to expand the image with the Page Up key.
We will now replace the ideal, inverting reflector at the right side (x = 0) with one
that partially reflects the wave incident from the left. Variable B provides scrollbar control
for Ry, the reflection coefficient for the dependent variable at that boundary, which is
restricted to real values between –1 and +1. Select variable B and vary it in “steps”
between those limits, to see a smooth transition from one ideal standing wave to the other.
The special case Ry = 0 (no reflection) separates the standing waves that have an imperfect
node at x = 0 from those that have an antinode there. The time dependence is sinusoidal
everywhere, and each frame in this moving picture shows a sinusoidal function of x. The
vertical scale is normalized to the amplitude of the incident wave.
Mm. 1. Lurching wave for Ry = +0.5 and its amplitude envelope (6 kb)
For typical “lurching” behavior, set B (Ry) equal to -0.5. Selecting variable C and
moving the slider in jumps displays the following features: C = 1 shows the total
waveform together with its stationary upper and lower amplitude envelope curves. The
latter show the maximum excursions of y at each value of x. C = 2 also marks each crest
with a small dot and plots its trajectory as a broken curve. C = 3 includes an open circle at
each temporal maximum, or point of contact with the upper amplitude envelope. Notice
that neighboring crests and temporal maxima take turns leading and following each other.
Setting C = 4 displays a graph of vf, the rightward velocity for each of those
features, plotted versus time. The dots for the crests and temporal maxima are the same as
those in the upper graph. Each vf is normalized to c, the speed of a single traveling wave.
These two curves have the same shape, but with a shift between them of T/4 where T is the
period. Return to the B variable (Ry) and watch what happens as you increase it by steps.
Everything simplifies at B = 0, with the two dots coinciding and advancing steadily. The
last few steps before B = 1 show how that ideal standing wave can be approached smoothly
via the lurching waves. If anything moves too quickly for you, slow it down using D.
3. Alternative representations
Open Mm. 2. The initial display again shows the lurching wave as the sum of two
traveling waves of (generally) unequal amplitudes. Select scrollbar variable A and jump its
value up to 1. The figure now shows an alternative decomposition of the lurching wave
into two standing waves. These are shifted from each other spatially by /4 and in time by
T/4. Select variable B (Ry) and vary it in steps. The standing wave components are seen in
isolation at B = 1. When B = 0 they have equal amplitudes and add up to the incident
wave. For the general case, the amplitude of the standing wave with an antinode at the
reflector is (1 + Ry) and that of the other one is (1 - Ry).
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Mm. 2. Standing wave components (6 kb)
Mm. 3. Complex exponential components (2 kb)
Set B = 0.5. Display both pairs of components simultaneously by selecting
variable A and jumping to A = 2. The standing waves are now shown in color.
Eliminate the traveling waves from the display by setting A = 3. Press the Up and
Left arrows on your keyboard simultaneously for about two seconds. This provides a new
perspective on what turns out to be a rotating three-dimensional figure. Prior to this we
have just been looking at its projection onto the x,y plane. (Using the Home key will
restore that projected view.) By varying the orientation in three dimensions, you can see
that the total wave has been turned into a wire figure and the two standing waves have
become rotating sets of paddles in orthogonal planes. The spokes of the blue and yellow
paddles represent vectors extending from the x axis to each standing wave. At each value
of x, the sum of the vectors for the two standing waves is the vector from the x axis to the
wire figure. (To display a few of the latter, set A = 4.) This can be interpreted as an
addition of phasors that represent two sinusoids of the same frequency with a phase shift of
 /2 between them. Setting D (speed) = 0 will freeze the figure in its position at t = 0 for a
systematic study of the effects of varying B (Ry). You can then use variable C to step the
figure through one cycle. With patient use of the direction arrows on the keyboard, you
may be able to orient the figure for a projection directly along the x axis, onto the y,z plane.
The paddles are then seen edge on, and the wire figure for the lurching wave projects as an
ellipse.
Each traveling wave used in the original superposition to obtain a lurching wave
can be considered as the projection of a helix. The sum of those helices is the same as the
rotating wire figure generated by the rotating sets of paddles. Open Mm. 3 to see this
representation. The initial display shows only the helix for the incident wave, which will
be shown consistently in red. It appears at first in an x,y projection. Use the direction
arrows as you did for Mm. 2 to reorient the figure in three dimensions. Setting A = 1
shows only the helix for the reflected wave. Notice how its solid color depends on the
magnitude of the reflection coefficient as you vary B (Ry). A = 2 shows both helices. Try
varying the surface transparency (an option in the Scrollbar menu). A fair amount of
transparency improves the x,y projection, but you’ll want little or none for an oblique view.
A = 3 includes the sum as a wire figure with spokes, and A = 4 shows only the sum. If you
have difficulty deciphering the three-dimensional image, expand it with Page Up and
freeze the rotation with D (speed) = 0. You can then use variable C to step through one
cycle.
In the efficient notation of complex exponentials, the helix for the incident
traveling wave is eikx-it and that for the reflected one is Rye-ikx-it. Their imaginary parts (z)
are eliminated by projection to leave just the real sinusoids. The sum can be expressed as
A(x)ei(x)e-it, where the amplitude A(x) is the radius of that figure as a function of x and
the phase delay (x) is the angle measured from the positive y axis (counterclockwise if the
positive x axis points toward you). The two curves of the amplitude envelope that were
shown in Mm. 1 are just  A(x). The x coordinates for the open circles that moved along
the upper envelope curve in Mm. 1 are the solutions of (x) -  t = 2 n.
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Up to this point in the discussion, energy has been lost only at the reflector (when
Ry < 1). We will now include the effects of viscous damping in propagation by using a
factor of e-ax for the incident wave and e+ax for the reflected one. This is equivalent to
letting k become complex. Open Mm. 4, select A (= a) for scrollbar control, and observe
the changes in the projected x,y view as you step this parameter from 0 to the maximum
value. Only the incident wave is shown initially, and it still has an amplitude of 1 at x = 0.
Leave A at its maximum value and vary B (Ry). Setting the latter to either extreme value
(+1 or –1) shows a regular progression from weak interference effects on the left to a
perfect antinode or node at x = 0. The strong lurching behavior on the right contrasts
noticeably with the smoother advance of the total wave on the left. Following a single zero
crossing from the far left to the right edge lets you focus on just the horizontal motion. The
complex figure in three dimensions is interesting, and you can probably anticipate its
shape. The value for the real part of k can be controlled with the C variable.
Mm. 4. Damping in propagation (2 kb)
Mm. 5. Complex reflection coefficients (3 kb)
A lightly damped lurching wave with a perfect node at one end is a fairly realistic
model for the standard classroom demonstration of standing waves on a rope. The nodes
away from the fixed end are not perfect, and the wave is never perfectly flat. Thus energy
supplied by a driver at the left end can be distributed to all the parts that are in motion,
where it compensates for the losses due to damping.
The termination at x = 0 that gives rise to the reflected wave has been purely
resistive so far--either a dashpot or a different medium that extends to infinity. An
additional benefit of the complex formalism is its convenience for representing a different
type of reflector, such as a mass or a spring, by letting Ry take on complex values. Open
Mm. 5 and expand it to fill the window using Page Up. Now the B variable controls just
the magnitude of the reflection coefficient, while C controls its phase, R. We have
returned to the condition a = 0 (no viscous damping). The plot on the right locates the
complex Ry within the unit circle. On the left, the basic shape of the amplitude envelope is
determined entirely by B (|Ry|). Changing the phase (C) from its initial value of 0 shifts the
antinode that had been at the origin to x =  R/2k. We now have a different way to change
Ry from +1 to –1, as suggested by the fact that the “paddles” for Ry = -1 can be obtained
from those for +1 by a shift of /4 in the +x direction and multiplication by i. Starting
from B = 1 and C = 0, just move counterclockwise on the unit circle by stepping C up to π.
The factor of i means that for Ry = -1, the waveform is flat at t = 0. This can be confirmed
by setting D (speed) = 0. Whenever the medium to the right of x = 0 is of finite extent, so
that waves interfere in that region, we will have use for this complex reflection coefficient.
4. Energy densities and intensity
Plots similar to those in the previous sections reveal some interesting details about the
distribution and transfer of energy in lurching waves. The physical example used here is
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that of an acoustic wave in a uniform pipe. We restrict the reflection coefficient to real
values, turn off the damping in propagation, and show just one wavelength on the screen.
Open Mm. 6 and expand it using Page Up. The plot on the left shows the acoustic
pressure p and its amplitude envelope for a lurching wave. Here we use B to vary R p, the
pressure reflection coefficient at x = 0. Its initial value is -0.5, representing a partial
reflection at an open end (with no end correction). The plot on the right shows the
potential energy density, which is proportional to p 2, along with its envelope. Setting A = 1
replaces p on the left with vx, the longitudinal particle velocity. Under the simplifying
assumptions of this section, Rv (the reflection coefficient for vx at x = 0) is just –Rp. Thus a
pressure node is an antinode for particle velocity and vice versa. On the right we now see
the kinetic energy density, which is proportional to vx2. The new plots are shown as dotted
lines to distinguish them from the earlier ones. Each dependent variable is normalized to
its maximum value for the incident wave, indicated by the tick mark on the vertical axis.
Mm. 6. Energy densities (5 kb)
Mm. 7. Lurching of energy density and intensity (4 kb)
Setting A = 2 shows both sets of curves. Notice that they coincide exactly in the
special case of a single traveling wave (Rp = 0). More generally a crest of p lines up with a
temporal maximum of vx, and vice versa. Thus the crests of these two variables take part
in a lurching race exactly like the one seen in Mm. 1. This is easy to observe when B =
 0.5. On the right we see the same race between the two energy densities.
Set A = 3 to see the total energy density and its envelope (solid curves) along with
the potential energy (dashed) and the kinetic energy (dotted) on the right. The tick mark on
the energy axis now represents the average value (either temporal or spatial) of the total
energy density for the incident wave. The pure traveling wave (Rp = 0) has its total energy
distributed in lumps that shift steadily to the right. Each ideal standing wave has a timeaveraged total density that is spatially uniform, but with a purely “sloshing” exchange
between the potential and kinetic parts and no net transport of energy. The general
lurching behavior is a temporal and spatial oscillation above and below an average value of
1 + Rp2, with twice as many repetitions per unit of time or distance as in the plot on the left.
Open Mm. 7. Use Page Up and select B (Rp). The upper plot shows the total
energy density, as in Mm. 6. The lower plot shows the instantaneous longitudinal intensity,
Ix = pvx. It also lurches, in this case with an average value of 1 - Rp2. The relationship
between these two curves is very simple for the single traveling wave, and it’s easy to
understand for the pure “sloshing” of either perfect standing wave. The x,y plots are again
projections of rotating figures in three dimensions. To clarify their relationship for the
general case, set D = 0 or a small value, point the x axes almost toward you, and vary B.
Except for their different average values, the two figures have identical shapes. One can be
obtained from the other by a shift of Δx = λ/8 and a counterclockwise rotation through π/2,
which amounts to a time delay of T/8. In a pure traveling wave, these spatial and temporal
shifts cancel each other. Jumping quickly between B = 0.5 and B = -0.5 for B illustrates the
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fact that these figures depend only on |R p|. If Rp > 0, the large variations in total energy
density at x = 0 are dominated by the potential energy, but at x = -λ/4 the kinetic energy
predominates. Those roles are reversed for Rp < 0, as can be seen in Mm. 6 but not here.
5. Summary and potential applications
Kinetic images used in an interactive mode provide opportunities for exploratory,
observational studies of wave kinematics. This qualitative and descriptive approach could
be valuable for students of engineering and science. A preliminary investigation in which
such elements as curiosity, intuition, and common sense are brought into play before the
relevant mathematics is introduced might encourage students to develop their own coherent
structure of understanding for the subject.
The nonstandard nature of the examples used here suggests that the availability of
this new graphics tool might change our ideas about what can be taught and at what level.
Clearly the material presented here is not suitable for use by students in this condensed
form. At the very least it needs to be fleshed out with the definitions of standard terms and
more discussion of the physics. As part of a larger effort to generalize the discussion of
waves, the topics covered here should be preceded by simple examples of superposition
using nonperiodic waves. An appropriate sequel would be the concept of input impedance,
or perhaps the superposition of sinusoids of different frequencies.
Another concern would be to decide how much mathematics should be presented
explicitly. What background can safely be assumed for the intended audience? Electronic
publication may alleviate that concern with the flexibility provided by internal links. The
pictorial, intuitive content could be the core from which individual readers branch out
according to their own estimates (or an instructor’s expectation) of what they are able to
handle. If they find the linked material to be intimidating, then they can back out of it,
knowing that it’s there when they are ready for it. A brief, early exposure to more
sophisticated mathematics may encourage the students to invest more effort in learning this
material when they encounter it later, since they will have already seen a use for it.
Acknowledgments
This work has been supported by the Paul S. Veneklasen Research Foundation and the
CSULB Scholarly and Creative Activities Committee. We are grateful to Mark
McLaughlin for his work on a preliminary study using MATLAB ®, which was presented at
the 17th International Congress on Acoustics, September 2001. David Parker provided very
helpful advice on the capabilities of DPGraph.
Appendix: DPGraph preliminaries
To install the viewer: 1. Click here to download the installer (395 kb), and save it onto
your desktop. When it has been saved, click on “Open” and then click “Yes.” This will
install the DP Graph Viewer (900 kb) in a few seconds. Then click “OK.” 2. Click on any
of the “Mm” kinetic graph links; the DPGraph Viewer will automatically open and display
the graph. 3. Use of the Windows Explorer Browser is recommended. Put that FIRST?
To vary parameters of the figure on display: Click on “Scrollbar” in the upper
tool bar. Select A, B, C, or D for the desired parameter, and then click “OK.” The slider in
the scrollbar on the right controls that parameter’s value, which is displayed in the box at
the bottom right. The role played by each parameter is shown in the box at the bottom left.
To move the slider in “steps,” click on an arrow directly above or below the scrollbar.
Click within the scrollbar above or below the slider to move it one “jump” (= 8 steps).
To rotate the graph, use the up, down, left, and right arrow keys on your keyboard.
The “Page Up” key zooms in; “Page Down” zooms out. To return the graph to its original
size and orientation, press the “Home” key. Other options can be found in the toolbar.
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