SI 2008:Study of Wave Motion

July 19, 2008

Martin BobbJoseph MarmersteinFeibi YuanCaden Ohlwiler

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Introduction

Goal -- Model Motion of Waves

MATLAB -- Programming Application

Method

Approximating the Wave Equation

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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

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Wave Equation

• This is a partial differential equation (college level math)

• Continuous • Approximated using the Finite Difference Method

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Boundary Conditions

• Behavior at the edges of the simulation

• Fixed– Out of phase reflection

• Free– In phase reflection

• No-reflection boundaries are very difficult

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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

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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

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Initial Disturbance

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Forcing Function

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Viscosity

• 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

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Low Viscosity

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High Viscosity

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Virtual Throat Simulation

• Two walls, space in between

• Three microphones– Behind, inside and in front of the throat

• Forcing function– 20 Hz

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Virtual Throat Visualization

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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 M

agnitude

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FFT on Microphone #2

Frequency (Hz)

Relative M

agnitude

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FFT on Microphone #3

Frequency (Hz)

Re

lative

Ma

gn

itud

e

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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”

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Virtual Room Visualization

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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

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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

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3D Waves from Single Initial Disturbance

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Corner Disturbance

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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

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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

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8 Disturbances

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3D Waves from Two Initial Disturbances

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Questions?