Visualizing Sound Propagation

Richard Batelaan Author

08/04/2020 Added

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Designing a visual model of sound propagation in a virtual room for educational use.

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[00:00:00.350]I am Richard Batelaan and I am an
[00:00:02.570]undergraduate physics and music major at
[00:00:04.702]the University of Nebraska-Lincoln.
[00:00:06.814]Today, I will be discussing my UCARE 2020
[00:00:09.344]project, Visualizing Sound Propagation.
[00:00:13.406]The goal of my project was to design a
[00:00:15.738]visual model of sound propagation in a
[00:00:17.908]virtual room for educational use.
[00:00:20.384]To accomplish this, I used Unity, which is
[00:00:23.074]a game engine that can be used for
[00:00:24.699]virtual reality. For my first approach, I
[00:00:28.079]tried to create an expanding sphere that
[00:00:30.068]could reflect off of virtual walls.
[00:00:32.853]Unfortunately, after the second reflection
[00:00:35.003]of the sound wave, the model becomes very
[00:00:38.115]unclear. For my second approach, I used a
[00:00:40.708]numerical solution to the 3D Wave Equation
[00:00:43.108]to show the points in space of highest
[00:00:45.139]sound pressure. This model clearly shows
[00:00:47.643]interference of sound waves after the
[00:00:49.468]first reflection. Here is a
[00:00:51.431]video of this model.
[00:01:05.303]The Expanding Ball Model shows the wave
[00:01:07.487]fronts of sound coming from a source.
[00:01:09.661]The room is part of a Revit model of the
[00:01:11.821]Peter Kiewitt Institute in Omaha. In the
[00:01:14.891]video, you can see the wave fronts change
[00:01:17.053]color after each collision and disappear
[00:01:19.186]after a while.
[00:01:58.225]I used four C# scripts to create my first
[00:02:01.415]model of sound propagation.
[00:02:03.277]The SpawnerSample script spawns spheres in
[00:02:05.736]every direction around a source, and each
[00:02:08.111]sphere is given a velocity away from that
[00:02:10.651]source. The effect is an expanding
[00:02:13.167]sphere, just like a wave front of sound.
[00:02:17.515]The BallBouncer script is attached to all
[00:02:19.923]of these spheres and makes them bounce off
[00:02:22.016]of the walls of the room. The script works by
[00:02:24.737]setting the angle of incidence
[00:02:26.262]equal to the angle of reflection at a collision.
[00:02:30.946]The Destroyer script is also attached to
[00:02:33.417]all of the spheres, and it destroys them
[00:02:35.809]based on number of collisions and the
[00:02:38.089]absorption coefficient of the material.
[00:02:40.639]The absorption coefficient is the
[00:02:42.466]percentage of sound absorbed and
[00:02:44.479]transmitted by the material.
[00:02:48.619]The ChangeColorOnCollision script changes
[00:02:51.523]the color of the spheres after each
[00:02:53.577]collision. After the first reflection,
[00:02:56.087]the color changes to black, then yellow,
[00:02:58.333]blue, red, and finally green.
[00:03:03.178]The Numerical Wave Equation Model shows
[00:03:05.808]the peaks of sound pressure when a source
[00:03:07.669]creates sound. The higher the sound
[00:03:09.866]pressure, the lower the transparency.
[00:03:12.576]Here are three views of the model, from
[00:03:14.656]the left, side, center, and right side of the room.
[00:03:30.215]Here is the 3D Wave Equation that I used
[00:03:32.585]for my model. The homogeneous part of the
[00:03:35.190]partial differential equation is:
[00:03:37.071]the second partial derivative of pressure
[00:03:39.296]with respect to time minus the speed of
[00:03:41.739]sound squared times the quantity,
[00:03:43.903]the second partial derivative with respect
[00:03:46.154]to x plus the second partial derivative
[00:03:48.111]with respect to y plus the second partial
[00:03:50.334]derivative with respect to z. The right
[00:03:53.254]side of the equation is the source
[00:03:54.874]function that I used for my model.
[00:03:57.214]Sine of omega t caused the
[00:03:58.614]source to oscillate over time.
[00:04:00.598]e to the negative alpha t caused the
[00:04:02.678]source to decay over time. And the last
[00:04:05.168]part of the source function causes it to
[00:04:07.824]have a spatial bell curve shape. There
[00:04:10.570]are many constants in the equation that I
[00:04:12.600]define later, like omega, which determines
[00:04:15.258]the frequency; alpha, which determines the
[00:04:17.865]source's rate of decay; and sigma, which
[00:04:20.515]is the standard deviation of the bell
[00:04:22.485]curve shape. Lastly, I set all of the
[00:04:25.345]initial and boundary conditions equal to
[00:04:27.421]zero because this is the simplest scenario.
[00:04:32.373]In order to solve my 3D Wave Equation, I
[00:04:34.713]used the Finite Difference Method to
[00:04:36.482]numerically approximate the solution. We
[00:04:39.001]use a simple rise over run approach to
[00:04:41.212]approximate all the partial derivatives
[00:04:43.158]of p. Using the same rise over run
[00:04:45.903]method, we can then approximate all the
[00:04:48.223]second partial derivatives of p, which
[00:04:50.593]simplifies to the following.
[00:04:53.427]When plugging these approximations into
[00:04:56.009]our 3D Wave Equation and solving for
[00:04:58.137]p(xi,yj,zk,th) we get a recursive formula.
[00:05:04.295]Starting with initial and boundary
[00:05:06.739]conditions, we can calculate any point,
[00:05:09.032]P(x,y,z, and t), using this formula.
[00:05:15.590]I used MATLAB to apply the Finite
[00:05:18.087]Difference Method to my 3D Wave Equation.
[00:05:20.634]The first part of this script defines all
[00:05:23.359]the variables. The first series of
[00:05:25.668]for loops assigns values to a
[00:05:27.488]four-dimensional array, x by y by z by t.
[00:05:31.668]The second series of for loops creates an
[00:05:34.387]animated 3D plot of a cross-section of
[00:05:37.087]the waves. The right video shown here is
[00:05:39.694]a cross-section at the top edge of the
[00:05:41.657]room. Notice that the waves start
[00:05:46.097]propagating late because the waves have
[00:05:48.327]not reached that position yet.
[00:05:54.827]The left video shown here is a
[00:05:57.060]cross-section at the center of the room.
[00:05:59.434]Notice that the waves start
[00:06:02.554]propagating immediately.
[00:06:10.511]The next section of code saved a video of
[00:06:14.201]these two animated 3D plots. The last
[00:06:17.114]series of for loops writes the data to
[00:06:19.394]64,000 different text files corresponding
[00:06:22.366]to a 3D position each having a list of one
[00:06:25.336]hundred time stamps for p.
[00:06:30.819]I used two C# scripts to show my model in
[00:06:34.205]Unity. The SpawnPixels script spawns
[00:06:37.428]cubes at every position in a 10x10x10 box,
[00:06:40.948]40 cubes in the x direction, 40 cubes in
[00:06:43.748]the y direction, and 40 cubes
[00:06:45.410]in the z direction, for a total of
[00:06:47.539]64,000 cubes, or pixels.
[00:06:51.695]The Wave script is attached to each pixel.
[00:06:54.502]It reads the text file corresponding to
[00:06:56.742]the position of that particular pixel to
[00:06:59.382]determine its transparency.
[00:07:03.115]In the future, we want this model to be
[00:07:05.375]adjustable to different timbres and
[00:07:07.604]frequencies. We will also work towards
[00:07:09.986]using different shapes of rooms instead
[00:07:12.136]of just a cubic box. Lastly, we can work
[00:07:15.411]towards giving this sound propagation
[00:07:17.173]model different boundary conditions,
[00:07:19.115]basically merging the pros of both models.
[00:07:24.718]And here are my references. Thank you!

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