Volume 33
http://acousticalsociety.org/
175th Meeting of the Acoustical Society of America Minneapolis, Minnesota 7-11 May 2018
Education in Acoustics: Paper 3aEDa3
Interactive (adjustable) plots and animations as teaching and learning tools DanielARussell
GraduatePrograminAcoustics,PennsylvaniaStateUniversity,UniversityPark,PA,16802;
[email protected] Interactive plots, such as are made possible with the Manipulate[ ] command in Mathematica, can be useful as teaching tools in the classroom, as well as learning aids for students outside of class. The use of adjustable sliders to change parameter values for an equation or system of equations allows for quick visual exploration of the effects of those changes on the resulting plot. Sometimes this visualization can help students understand difficult conceptual meaning hidden in mathematical expressions. Similarly, an adjustable interactive animation can effectively facilitate an understanding of concepts that are sometimes difficult to grasp from words or equations, or even ”fixed” animations. This talk will demonstrate the creation of interactive plots and animations, and showcase several examples which the author has developed and used for teaching acoustics at the graduate and undergraduate levels. In addition, we will also discuss ways that students can be encouraged to create their own interactive plots or animations to aid in their own understanding.
PublishedbytheAcousticalSocietyofAmerica
1. CREATING INTERACTIVE ANIMATIONS WITH MATHEMATICA A. BACKGROUND – ACOUSTICS AND VIBRATION ANIMATIONS WEBSITE I started making animations of plots and oscillating systems in 1992 while in graduate school working toward the Ph.D. in Acoustics at Penn State. My Ph.D. advisor, Dr. Victor Sparrow, used Mathematica to created movies of oscillating systems and wave motion, and he demonstrated these animations in the courses he taught. Vic shared his ”tricks” with me and encouraged me to design and create my own animations. In the process of writing this POMA paper, I discovered a cache of his old animations,1 and have converted them from Quicktime movies into animated GIF files.2 My own initial purposes for creating animations were two-fold. First, I simply wanted to better understand the acoustics I was learning; I would reproduce plots in textbooks and explore how those plots changed when the parameters were varied. Combining a sequence of such plots into an animation helped me visualize what equations ”looked like” and how a result depended on its variables. Secondly, there were some concepts involved with my dissertation that could be explained so much easier with an animation than I could express verbally or in writing. After securing a faculty position, I continued to create animations to help me explain waves and vibration concepts in my own teaching. Sometime around 1997, I began posting animations to a website3 for others to use. The website has become quite popular, averaging between 8,000 and 10,000 hits per month, and has been the topic of several papers at previous ASA meetings.4–7 In this paper I will discuss some recent things I’ve been doing with interactive (adjustable) animations and plots along with some of the challenges I have encountered regarding their use and dissemination. B.
USING THE MANIPULATE[ ] AND ANIMATE[ ] COMMANDS
Version 7.0 of Mathematica (as of this writing, the current version is 11.3) introduced two functions which greatly simplified the creation of adjustable animations and interactive graphics. The first function, Animate[ ] allows one to create an automatically running animated output; the second function, Manipulate[ ] allows for the manual adjustment of variable parameters through the use of sliders and buttons. A couple of years later, with version 8.0, Wolfram introduced the Computable Document Format (CDF) which allowed one to save an interactive (adjustable) animations to a standalone file or to embed it into the HTML code on a webpage.8 The end user could download a free CDF player9 enabling interaction with the animation without having to purchase the full version of Mathematica. A simple example of an interactive (adjustable) animation using the Animate[ ] and Manipulate[ ] commands is shown in Fig. 1. In this example, a sinusoidal wave function p(x, t) = A sin(ωt − kx)
(1)
is plotted as a function of position, with time continuously increasing so that the wave function moves to the right. The sliders allow the user to adjust the amplitude A, wavenumber k, and angular frequency ω. Students can explore the fact that changing k causes the wavelength to change, and keeping k fixed while changing ω (or vice versa) causes the propagation speed to change, since c = ω/k. The Mathematica code to produce this animation is:
Figure 1: Simple example of an adjustable animation of a traveling wave function. The animation runs with time; moving the sliders adjusts the amplitude, the wavenumber (wavelength) and frequency. Link to the CDF for this animation: https://tinyurl.com/y8y8xwuk. Manipulate[ Animate[ Plot[ A Sin[w t - k x], {x,0,10}, PlotRange->{-1.1,1.1}], {t,0, Infinity}], {w,1,5}, {k,1,5}, {A,0.1,1}]
The middle line of this code plots Eq. (1) as a function of position over the range {x, 0, 10} with a vertical scale ranging from −1.1 to +1.1. The lines of code immediately above and below animate this plot as a function of time; the upper time limit of ”Infinity” keeps the plot running continuously unless the user clicks on the play/pause button. The outermost two lines of code create the sliders allowing manual adjustment of the variables ω, k, and A. The Computational Document Format (CDF) file for this, and all animations described in this paper, may be downloaded using the links in the figure captions. The CDF interactive animations may be viewed with the free CDF player from Wolfram.com.
2. WAYS TO INTERACT WITH ANIMATIONS AND PLOTS A. CHANGING VARIABLES WITH SLIDERS When using Manipulate[ ] to create an interactive plot, the default means of changing a variable is a slider, as was shown in Fig. 1. Another example using sliders is shown in Fig. 2, which plots the far-field directivity pattern radiated by a circular baffled piston, as given by H(θ) =
2J1 (ka sin θ) . ka sin θ
(2)
Actually, the directivity is plotted as 20 log H(θ) as a sound pressure level. The user can move the slider to change the value of ka to observe how the directivity pattern changes. When ka is small, the baffled circular piston is omnidirectional. Increasing the value of ka (higher frequency or larger piston) causes the radiation pattern to become more directional along the central axis, with side lobes appearing. The alternating colors of blue and yellow indicate the phase reversal (positive and negative) of the neighboring lobes.
As an aside note, the yellow-blue color scheme used in Fig. 2 was chosen to mimic the recently deleloped colormap cividis, which has been optimized to be perceptually uniform in hue and brightness and avoids the use of reds and greens.10 This colormap is supposed to be optimal in the sense that it is interpreted the same way by people with or without color vision deficiency. This colormap has been implemented in COMSOL11 and MATLAB12 but has not yet been implemented for Mathematica. There is an online site where you can upload an image to be tested for various types of color blindness.13 A slightly more complicated interactive plot using sliders is shown in Fig. 3. In the secondsemester graduate-level theoretical acoustics course at Penn State, we discuss the radiation from baffled pistons using the Rayleigh Integral and the spatial Fourier transform, which converts the spatial velocity profile of the baffled source into wavenumber space. The Rayleigh integral may be expressed as ! ! −iωρo eikr ∞ ∞ pˆ(⃗r ) = uˆn (xs , ys ) e−ikx xs e−iky ys dxs dys (3) 2π r −∞ −∞ The surface velocity distribution uˆn (xs , ys ) for a rectangular piston may be expressed in terms of the “rect” function, for which the spatial Fourier transform is a sinc function, " # $ % $ % (4) F rect(xs /Lx ) rect(ys /Ly ) = Lx Ly sinc kx Lx /2 sinc ky Ly /2 .
The x and y components of the wavenmuber may be expressed in terms of spherical coordinate direction angles as kx = k sin θ cos φ and ky = k sin θ sin φ so the radiated pressure becomes & ' & ' iωρc 1 1 uˆn Lx Ly sinc kLx sin θ cos φ sinc kLy sin θ sin φ . (5) pˆ(r, θ, φ) = − 2π 2 2
Figure 3 shows the effect of adjusting the dimensions, Lx and Ly , of the rectangular piston to illustrate the radiation directivity for a point piston, a thin line piston, and a large square piston.
Figure 2: Showing the effect on the directivity pattern for a baffled circular piston, using a slider to change the value of ka. (left) Directivity is omnidirectional for ka = 1, but becomes very directional (middle) when ka = 11. (right) the image for ka = 11 before it has fully rendered. Link to the CDF for this animation: https://tinyurl.com/yac9vdvp.
Figure 3: The effect of source dimensions of a baffled rectangular piston, on the resulting wavenumber space and angular directivity function. In each plot, the left image shows the dimensions of the piston, the middle plot shows the k-space plot, and the right plot shows the angular directivity. The users controls the size and shape of the rectangular piston by moving the Lx and Ly sliders. Link to the CDF for this animation: https://tinyurl.com/ybo8kd3k.
As the sliders are moved, the adjustable graphic shows the shape of the piston, the corresponding spatial Fourier transform in k-space, and the resulting angular directivity function. B.
CHANGING INTEGER VALUES WITH BUTTONS
While sliders allow a user to change a variable over a continuous range, there are some cases where discrete integer values may be more appropriate. For such cases, one can use radio-style buttons by using the modifier ControlType → SetterBar within the Manipulate[ ] command. Figure 4 uses two sets of buttons with integer values to select the (n,m) indices for a graphical display of the spherical harmonics: ( (2n + 1) (n − m)! m Ynm (θ, φ) = P (cos θ) ejmφ . (6) 4π (n + m)! n The user simply clicks on the values of n and m corresponding to the desired spherical harmonic and the real and imaginary parts of the spherical harmonic are displayed. Since m ≤ n, nothing is displayed if m > n.
Figure 4: Radio buttons are used to select the integer indices (n,m) for the various combinations of the spherical harmonics. Link to the CDF for this animation: https://tinyurl.com/y8cbvvx6.
Figure 5 shows another example how radio buttons may be combined with a time-running animation to allow a user to select between the three natural modes of oscillation for a 3-DOF mass-spring system.14 As was the case for the example in Fig. 1, the Animate[ . . . , {t,0,Infinity}] command embedded within Manipulate[ ] causes the animation to run continuously. When the user selects mode 1, all three masses move in the same direction, but with the middle mass moving with slightly larger amplitude. When button 2 is clicked, the middle mass is stationary (a node) while the outer two masses oscillate asymmetrically in opposite directions. For mode 3, the middle mass moves in one direction while the outer two masses move together in the opposite direction. C. COMBINING SLIDERS AND RADIO BUTTONS There are wave phenomena for which it is desirable to create an animation using both sliders and radio buttons, along with a continuously running animation. Such an example, shown in Fig. 6, illustrates the behavior of evanescent modes in a waveguide. The author uses this animation
Figure 5: Using radio buttons to select between the three normal modes of oscillation for a 3-DOF mass-spring system. Website containing these animations: https://tinyurl.com/yat8jah6.
as a prelude to a physical demonstration of evanescent modes in a rectangular waveguide.15 The animation in Fig. 6 is continuously running with time; the piston at the left moves back and forth and when the mode 0 button is selected, as shown in case (a), a plane wave propagates down the waveguide to the right. Since a plane wave will always propagate down a rigid-walled waveguide at any frequency, moving the frequency slider simply changes the frequency with which the piston oscillates and the wavelength of the plane wave propagating down the waveguide. In case (b) the mode 1 button has been selected, but the frequency slider is set to a value below the (normalized) cut-on frequency for that mode. When driven below cut-on, this non plane-wave mode is evanescent; it does not propagate, but exponentially decays with distance down the waveguide. In case (c) the frequency is increased to slightly above the (normalized) cut-on value, and mode 1 now propagates down the waveguide. Case (d) shows mode 1 propagating at a higher frequency, but below the cut-on frequency for mode 2. Case (e) shows that if the frequency is not changed, but mode 2 is selected, this mode will is evanescent and does not propagate below its cut-on frequency. In case (f) the frequency for mode 2 is increased to a value above its cut-on frequency and this mode now propagates. An animation like this can be very useful for helping students understand the behavior of evanescent modes and cut-on frequencies in a waveguide.
(a)
(b)
(c)
(d)
(e)
(f)
Figure 6: Using a slider along with radio buttons (and a continuously running animation) to illustrate how an acoustic mode in a waveguide will be evanescent if driven below the cut-on frequency, while propagating down the waveguide when driven above the cut-on frequency. Link to CDF file for this animation: https://tinyurl.com/ycodqcok.
3. ADJUSTABLE ANIMATIONS AND INTERACTIVE GRAPHS A. AN EXAMPLE OF AN ADJUSTABLE ANIMATION – CRANKSHAFT As an additional example of an animation for which real-time adjustment of an important parameter allows for greater understanding, consider the crankshaft, shown in Fig. 7. The wheel rotates counterclockwise with constant angular velocity, while the piston is constrained to oscillate back and forth along the horizontal axis. The crankshaft is an example of a simple system that undergoes non-harmonic periodic oscillation.16 The relationship between the wheel radius a, rotational speed ω, shaft length L, and horizontal piston location x maybe derived from the geometry of the problem as L2 = a2 + x2 − 2ax cos ωt . (7) The position of the end of the shaft may be solved as ) x(t) = a cos ωt − L2 − a2 sin2 ωt ,
(8)
which is not sinusoidal motion, unless L ≫ a at which point x(t) ≈ a cos ωt.
𝜃
𝜃 = 𝜔t
.
a
𝜔
x
Figure 7: A crankshaft is an oscillating system. Figure 8 shows two stills from an interactive animation of the crankshaft. While the animation is running, the wheel rotates, the crankshaft oscillates back and forth horizontally, and the plot traces out the time history of the crankshaft position. The slider controls the shaft length and as the slider is moved, the plot and the animation both adjust to show the correct motion and time history. The transition from sinusoidal to non-sinusoidal is clearly observed as shaft length is shortened.
Figure 8: A crankshaft is a non-harmonic oscillating system if the shaft length is small, but approximates to a sinusoidal oscillation if the shaft length is long. Link to webpage with animations and source code: https://tinyurl.com/yc6wuwqp.
B.
AN EXAMPLE OF AN INTERACTIVE GRAPH – ABSORPTION IN AIR
The topic of absorption of sound waves in air is important for discussions of sound propagation outdoors and for many noise control applications. The attenuation of sound as it travels through air is often reported as an absorption coefficient α with units of dB/m (or sometimes dB/10m or dB/100m). The details are a bit complicated because air is a polyatomic gas. The absorption of sound energy in air depends strongly on the frequency of the sound wave as it excites vibrational modes of the diatomic molecules of nitrogen and oxygen, and the amount of absorption at a given frequency is strongly affected by the concentration of water vapor molecules. In many textbooks, the phenomenon of molecular relaxation is described conceptually and the value of the attenuation coefficient is estimated from a graph of α versus frequency, such as is shown in Fig. 9. This graph includes curves for three values of the relative humidity (0%, 10%, and 100%) all at a temperature of 20◦ C and at an ambient pressure of one atmosphere.17 Aside from the fact that attempting to extract a value of the absorption coefficient α from the graph in Fig. 9 will provide only an approximate value, what does one do if you needed the absorption coefficient for a temperature other than 20◦ ? Or for a humidity index other than 0%, 10%, or 100%? Or, at a physical location where the atmospheric pressure is significantly different from the value at sea level? Of course, you could calculate the attenuation coefficient α directly from the equations provided in the literature according to18, 19 α = αclassical + αN2 + αO2 ,
(9)
where αclassical = α N2 = αO 2 =
2
f 1.84 × 10
'−5/2 *
&
T T0
'1/2
,
(10)
+ e−3352.0/T f 0.1068 , Fr,N + f 2 /Fr,N & '−5/2 * + T e−2239.1/T 2 f 0.01275 , T0 Fr,O + f 2 /Fr,O 2
&
−11
T T0
(11) (12)
with the oxygen and nitrogen relaxation frequencies given by '& ' & pa 4 0.02 + h 24 + 4.04 × 10 h , Fr,O = patm 0.391 + h & '& '−1/2 * + pa T −4.17((T /T0 )−1/3 −1) Fr,N = 9 + 280 h e , patm T0 along with the definitions & ' psat h = hrel pa
and
log10
&
psat patm
'
&
T01 = −6.8346 T
'1.261
+ 4.6151 ,
(13) (14)
(15)
and the ambient values T0 = 293.15 K(20◦ C), T01 = 2.73.16 K, patm = 1.01325 × 105 Pa. In all of these equations, T is the temperature in Kelvin, f is frequency in Hz, pa is the local ambient pressure, and hrel is the % relative humidity. One could argue that it is good practice for students
Figure 9: Plot of the absorption coefficient for air, α (dB/m), as a function of frequency for various values of the relative humidity and at a temperature of 20◦ C and at an ambient pressure of 1 atm, as printed in a popular acoustics textbook.17 to go through the experience of performing these calculations to obtain an answer. But, this set of equation does not make observing the dependence of absorption on humidity, temperature, or pressure an easy endeavor. In order to more fully explore the dependence of absorption on these variables, one could use an interactive plot as shown in Fig. 10. Here, the absorption coefficient is plotted as a function of frequency and the plot updates as the user moves sliders to adjust the values of the relative humidity, temperature, ambient pressure. The slider for frequency causes a pair of green lines to pinpoint the value of absorption on the graph, and the numerical value of the absorption coefficient (db/100m) for the desired frequency is shown in the blue box at the top left of the plot. The version of the interactive plot with sliders (on the left of Fig. 10) has some pedagogical advantages. In addition
Figure 10: An interactive plot that calculates the absorption coefficient (dB/100m) for a specified % relative humidity, frequency, temperature (◦ C), and ambient pressure (atm). The version on the left uses sliders to adjust the parameters; the version on the right allows the user to enter values directly. Link to the CDF for the slider version of this interactive plot: https://tinyurl.com/y7fuofnh to the classical and total absorption curves, the plots also show the oxygen (blue) and nitrogen (red) absorption curves. Moving the frequency slider allows the user to see how the absorption curve changes as the humidity, temperature, and pressure are varied. It is especially interesting to watch the absorption curve vary with humidity; one can observe the absorption values at a specific frequency increase, pass through a maximum value, and decrease as the humidity increases from 0% to 100%. An interactive plot like this allows the user to extract a numerical value, but it also allows for a graphical exploration of a complicated system of equations in an accessible manner. However, if the user is primarily interested simply obtaining numerical values for the absorption coefficient, then the version on the right side of Fig. 10 might be more appropriate; this version
allows the user to enter the values directly. Since plot uses a log-log scale, selecting the frequency using a slider limits the resolution and the ease of obtaining a specific frequency. The numerical entry version is easier for simply and quickly obtaining a value of the absorption.
4. QUESTIONS ABOUT USING INTERACTIVE ANIMATIONS A. WHAT ARE INTERACTIVE ANIMATIONS USEFUL FOR? This author primarily uses the interactive animations I have created for two purposes. First, creating an interactive animation helps me understand a concept or phenomena better so that I can more clearly explain it when I teach. Second, using the interactive animation during a lecture can also help students see how the equations or plots results depend on the variables in question. Others papers in this session21, 22 have demonstrated that interactive physics simulations20 can be effective tools for pre-lecture or pre-lab exercises to help gain some familiarity with the subject matter before formally interacting with the concepts during class. Or, interactive simulations may be utilized for in-class group work or post-lecture exploration to enhance understanding of a topic.23, 24 Making interactive animations and adjustable plots available for other educators and students to use would allow a variety of uses and applications. B.
ARE INTERACTIVE ANIMATIONS BETTER THAN STATIC ANIMATIONS?
The examples discussed in this paper are interactive animations and adjustable plots with which a user interacts by moving a slider, clicking a button, or entering a number. However, there are some cases where an interactive animation might not be the best option. The 3D directivity plots shown in Fig. 2 take a couple of seconds to fully render; a partially rendered plot is shown Fig. 2(right). The interactive plot shown in Fig. 10 takes a second or two to perform all of the calculations in Eqs. (9)-(15) and to update the plot. The ability to move a slider and watch the directivity plots change may be an acceptable trade-off to the time required for rendering, but when the rendering or computation time is much slower than the user can move the slider, the interaction loses some of its immediate usefulness, if the purpose is to show dependence on a single variable. If interaction is not essential to the purpose of the animation or plot, it may be better to just create a smoothly playing movie showing how the plot or the motion changes as a single parameter is varied. Several years ago, I was developing an interactive animation to compare the scattering as a function of frequency from rigid spheres and cylinders, and from rigid and pressure release spheres. The calculation for the scattering of a plane wave from a rigid sphere involves taking the magnitude of a sum of a product of Legendre polynomials with spherical Bessel functions and Hankel functions and their derivatives, N e−jkr , jn ′ (ka) pscat (r, θ) = −jPo (2n + 1) (2) ′ Pn (cos θ) , kr n=0 hn (ka)
(16)
where the number of terms N in the sum must be at least twice the largest ka value of interest. The computation time – to calculate the magnitude of a sum of 40 terms involving products of several complex special functions and then to render an updated plot – was long enough that moving the slider and having to wait for the simulation to catch up became frustrating. The goal of the
animation was to show how the scattered pressure field changed with frequency while comparing the results for two different objects. This purpose did not explicitly require interaction; it could be accomplished with a movie showing how the plots changed with increasing frequency. So, instead of using the adjustable plots, I generated a large collection of still frames and combined them into a smoothly varying animated GIF movie file.25 Figure 11 shows two still frames from one of these animated movies. In this case, the smoothly playing “static” animation was more useful than an adjustable plot with a moveable slider and a long time-lag for the graphics to catch up.
Figure 11: Still frames from a non-interactive animated movie that smoothly shows the effect of changing the frequency ka on the scattering of a plane wave from a rigid sphere compared to a rigid cylinder. The animated movie runs smoothly without any lag or rendering issues that are inherent to the interactive animation used to produce the images. Link to animation: https://tinyurl.com/y9m6xnpe
C. WHAT IS THE TRADEOFF BETWEEN USING AND CREATING? Another paper presented in this education session discussed the trade off between having undergraduate students simply use a black-box computer software package, like MATLAB with SimulLink and COMSOL, to explore complicated mechanics and acoustics problems versus requiring students learn the software well enough to be able to develop their own simulations.26 The question doesn’t have a simple answer as there are educational benefits for both approaches. One could ask the same question about the interactive animations discussed in this paper. Is it preferable for a teacher or student to interact with an adjustable plot or animation for a few minutes, or is it better to have students learn to make their own interactive animations? Some of these animations, like that in Figs. 8 or 4 took roughly a full day, from conception to final webpage posting. Others, like Figs. 3, 6, and 10 required many hours over several days before they reached final form, even having a pretty good familiarity with Mathematica. If the goal is to help students learn acoustics, then interacting with an animation that someone else created meets the need. Making plots, as was my initial experience as a grad student, is a worthwhile experience. But at some point the process of producing high quality, visually pleasing the mathematically and physically correct, interactive animations can be as much a lesson in learning how to code as it is about learning and understanding acoustics. D. HOW TO BEST SHARE THESE INTERACTIVE ANIMATIONS? When Wolfram first introduced the Computable Document Format (CDF), the free CDF player was compatible with several popular web browsers, and interactive CDF simulations could be
embedded into webpages, making it very easy for users to interact with them. Wolfram had also expressed plans to make the CDF player available for mobile devices. This author’s original intent was to create a large number of interactive CDF animations and imbed them into webpages for others to use. Embedding interactive CDF animations in a webpage would allow for some text explaining the background theory and context, along with instructions on how to interact with the animation, However, CDF files are are no longer compatible with any web browser and Wolfram has no intentions to fix this problem. At this point in time, CDF animations are only accessible as a separate downloadable CDF file, viewable with the free Wolfram CDF Player.9 At present, Wolfram’s preferred method of dissemination is to post demonstrations on the Wolfram Cloud demonstration website (www.wolframcloud.com) where visitors can interact with and run animations. However, this requires that the CDF creator purchase (with real money) “cloud points” which are deducted each time a visitor interacts with the demonstration. Given the number of visitors (8,000-10,000 per month) to my current website,3 this is not a financially viable option. At the present moment, it appears that the simplest way to share my interactive CDF animations is to create a webpage showing some ‘static’ animated movies, with a link to the standalone interactive CDF file. Interested users could then download and interact with the CDF files using the free Wolfram CDF Player.9
5. SUMMARY This paper has provided several examples of interactive animations and adjustable plots, which were created in Mathematica using the Manipulate[ ] and Animate[ ] commands. A combination of different methods of allowing interaction were illustrated for a variety of acoustic and vibration phenomena. All of the animations discussed in this paper may be downloaded as standalone CDF files by following the links in the figure captions. Full Mathematica source code will be shared by the author upon request.
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