Wave Phenomena

Professor Mark F. Hamilton
M E 384N, Spring 2024
Walker Department of Mechanical Engineering
University of Texas at Austin

Contents

1. Mathematical preliminary
• Linear systems theory
• 1D Fourier transform
• 2D spatial Fourier transform
• Transform theorems
• The angular spectrum
• Hankel transform
• Dirac delta functions
2. Fourier acoustics
• Pressure source
• Velocity source
• Intensity
• Code
• Radiation from plate due to a 1D bending wave
• Generalization to elastic waves in isotropic solids
• Nearfield acoustical holography (NAH)
3. Rayleigh integrals
• Angular spectrum of a spherical wave
• Derivation of the first Rayleigh integral using Fourier acoustics
• Derivation of the second Rayleigh integral using Fourier acoustics
4. Fraunhofer approximation
• Derivation
• Beamwidth
• Beam steering
• Reflection problems
• Arrays
5. Fresnel approximation
• Focused sources
• Fresnel approximation
• Far field of Fresnel approximation
• Gaussian beams
• Circular piston
• Rectangular piston
• Derivation of parabolic equation
6. Moving media
• Galilean transformation of wave equation
• Doppler effect
• Sound in ducts with uniform flow
• Refraction with flow
• Bending wave in flow
• Piston radiation into crossflow
7. Periodic media
• Mass-spring transmission line
• Waves in layered media
8. Scattering
• General definition of Green's function
• Green's functions of inhomogeneous 1D, 2D, 3D Helmholtz equations
• Multipole expansion
• Reciprocity
• Helmholtz-Kirchhoff Integral
• Scattering from inhomogeneities
• Fraunhofer, Born, and Rayleigh scattering
• Born scattering from bubbles below resonance
• Scattering from bubbles at arbitrary frequencies

Additional handouts

• Delta function identities
• Fourier and Hankel transform identities
• List of Fourier transforms
• List of Hankel transforms

About

These are my class notes from Dr. Hamilton's lectures. My notes almost exactly follow the material presented in the lecture (aside from a few topics that have been re-arranged for conceptual consistency), and I have made some additional comments and linked references in deep blue. Also, the relevant quotations of famous physicists at the tops of the pages were my addition. Please send suggestions and corrections to:

Chirag Gokani
chiragokani [at] gmail.com
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