SI 2008: Study of Wave Motion July 19, 2008 Martin Bobb Joseph Marmerstein Feibi Yuan Caden Ohlwiler

2 EMPOWER. PARTNER. LEAD. Introduction  Goal -- Model Motion of Waves  MATLAB -- Programming Application  Method  Approximating the Wave Equation

3 EMPOWER. PARTNER. LEAD. MATLAB Software with simulation and visualization functions “Matrix Laboratory” –Uses matrices to perform complex calculations Most of our group has limited programming experience –No previous MATLAB experience

4 EMPOWER. PARTNER. LEAD. Wave Equation This is a partial differential equation (college level math) Continuous Approximated using the Finite Difference Method

5 EMPOWER. PARTNER. LEAD. Boundary Conditions Behavior at the edges of the simulation Fixed –Out of phase reflection Free –In phase reflection No-reflection boundaries are very difficult

6 EMPOWER. PARTNER. LEAD. Two Dimensional Waves Wave in x-y plane –Like the surface of a lake –Third dimension represents wave height Set conditions to represent air –Wave propagation speed –Viscosity Nyquist frequency –Highest frequency that can be resolved accurately –Simulation must run at twice that –22100 Hz

7 EMPOWER. PARTNER. LEAD. Initial Disturbances and Forcing Functions Initial disturbance –One disturbance, dissipates over time (much like a shock wave) Forcing Function –Continuous Output –Similar to a speaker playing a single note –Follows the path of a sine wave

8 EMPOWER. PARTNER. LEAD. Initial Disturbance

EMPOWER. PARTNER. LEAD. Forcing Function

10 EMPOWER. PARTNER. LEAD. Viscosity –Measure of the resistance to motion from a fluid –Adds a new term to wave equation Makes the wave equation more realistic Molasses vs. Water

EMPOWER. PARTNER. LEAD. Low Viscosity

EMPOWER. PARTNER. LEAD. High Viscosity

13 EMPOWER. PARTNER. LEAD. Virtual Throat Simulation Two walls, space in between Three microphones –Behind, inside and in front of the throat Forcing function –20 Hz

EMPOWER. PARTNER. LEAD. Virtual Throat Visualization

EMPOWER. PARTNER. LEAD. Fast Fourier Transforms (FFTs) Fourier Transform –Mathematical method –Takes a function in time and changes it to frequency components –Shows the frequencies at which a system is responding Fast Fourier Transform –Numerical method –Computes the Fourier Transform quickly

16 FFT On Microphone #1 Frequency (Hz)‏ Relative Magnitude

17 FFT on Microphone #2 Frequency (Hz)‏ Relative Magnitude

18 FFT on Microphone #3 Frequency (Hz)‏ Relative Magnitude

19 EMPOWER. PARTNER. LEAD. Virtual Room Simulation Based on geometry of BALE Gallery room Forcing Function of 110 Hz –“A” string on a guitar Three microphones –Behind column, in front of column, behind “speakers” Simulation was not entirely successful –Solution tends to “ring”

EMPOWER. PARTNER. LEAD. Virtual Room Visualization

21 EMPOWER. PARTNER. LEAD. Three Dimensional Waves Volume divided into finite pieces Wave “height” represents compression instead of actual wave height Requires more computing power –Simulations take more time –Can take hours Waves dissipate much faster than in 2D –Forcing functions have a less noticeable effect

EMPOWER. PARTNER. LEAD. Three Dimensional Visualization Tried multiple techniques in MATLAB –Isosurfaces Generates meshes at specific wave heights Did not provide relevant visual –Volume Rendering Passes light through the simulation Used in 3D visualizations Alpha mapping –Does not display wave heights close to zero –Makes them transparent –White space isn’t necessarily motionless

EMPOWER. PARTNER. LEAD. 3D Waves from Single Initial Disturbance

EMPOWER. PARTNER. LEAD. Corner Disturbance

EMPOWER. PARTNER. LEAD. Conclusions Learned about wave motion –Computer simulations are difficult to make exact –Everything is approximated Learned how to program in MATLAB –Learned about simulation and visualization Learned how to use Linux

26 EMPOWER. PARTNER. LEAD. Acknowledgments Troy Baer -- Project Leader Armen Ezekielian -- MATLAB support Elaine Pritchard -- Food/snacks/organization Daniel & Brianna -- Dorm supervisors All the staff that gave presentations James Rader -- Gallery blueprints Third-party MATLAB add-ons: –Blinkdagger.com – positiveFFT function –Mathworks.com -- vol3d function

EMPOWER. PARTNER. LEAD. 8 Disturbances

EMPOWER. PARTNER. LEAD. 3D Waves from Two Initial Disturbances

EMPOWER. PARTNER. LEAD. Questions?