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ADVANCED STRUCTURAL DYNAMICS AND ACOUSTICS13.811 Lecture 3
13.811Advanced Structural
Dynamics and Acoustics
Lecture 3
ADVANCED STRUCTURAL DYNAMICS AND ACOUSTICS13.811 Lecture 3
The Acoustic Wave Equation
1-dimensional propagation
Plane Waves
Fourier Transform
Acoustic WavenumberForward
PropagatingBackward
Propagating
..
ADVANCED STRUCTURAL DYNAMICS AND ACOUSTICS13.811 Lecture 3
The Acoustic Wave Equation
1 D Plane Wave
Field Parameters
AcousticImpedance
Particle Displacement
Particle Velocity
Acoustic Pressure
ADVANCED STRUCTURAL DYNAMICS AND ACOUSTICS13.811 Lecture 3
3D Acoustic Plane Waves
θφ
R
z
x
y
VS 2D Fourier Transform
Wavefronts
k r = const
k
not independent
Wavevector – Propagation Direction
Vertical Wavenumber - Dependent Variable
Depth-separated Helmholtz Equation
Vertical Wavenumber
Radial Wavenumber
ADVANCED STRUCTURAL DYNAMICS AND ACOUSTICS13.811 Lecture 3
3D Acoustic Plane Waves
2D Fourier Transform
Depth-separated Helmholtz Equation
Vertical Wavenumber
Radiation Condition
k
kx
kz
k = k sin θxk = k cosθz
ADVANCED STRUCTURAL DYNAMICS AND ACOUSTICS13.811 Lecture 3
3D Acoustic Plane Waves
2D Fourier Transform
Depth-separated Helmholtz Equation
Vertical Wavenumber
>
Radiation Condition
Grazing Plane Wave
Evanescent Wave
ADVANCED STRUCTURAL DYNAMICS AND ACOUSTICS13.811 Lecture 3
3D Acoustic Plane Waves
Radial Wavenumber
Radiating Waves
Evanescent Waves
Vertical Wavenumber
Wavefield Extrapolation
Wavefield Extrapolation
At each radial wavenumber the field at any other depth z is determined by multiplication by a simple exponential
Propagation Angles
ADVANCED STRUCTURAL DYNAMICS AND ACOUSTICS13.811 Lecture 3
Radiation From Infinite PlateDepth-separated Solution
Plate Vibration
Boundary Condition
Vertical Displacement
Image removed due to copyright considerations. See Figure 2.10 in Williams, E. G. Fourier Acoustics. London: Academic Press, 1999
Image removed due to copyright considerations. See Figure 2.10 in Williams, E. G. Fourier Acoustics. London: Academic Press, 1999
ADVANCED STRUCTURAL DYNAMICS AND ACOUSTICS13.811 Lecture 3
Radiation From Infinite PlateRadiation Regime
Plate Vibration
Radiated Field
ADVANCED STRUCTURAL DYNAMICS AND ACOUSTICS13.811 Lecture 3
Radiation From Infinite PlateEvanescent Regime
Plate Vibration
Evanescent Field
>
Grazing Plane Wave
Evanescent Wave
ADVANCED STRUCTURAL DYNAMICS AND ACOUSTICS13.811 Lecture 3
3D Radiation From Infinite PlateRadiation Regime
ADVANCED STRUCTURAL DYNAMICS AND ACOUSTICS13.811 Lecture 3
3D Radiation From Infinite PlateEvanescent Regime
Image removed due to copyright considerations. See Figure 2.10 in [Williams].Image removed due to copyright considerations. See Figure 2.10 in [Williams].
ADVANCED STRUCTURAL DYNAMICS AND ACOUSTICS13.811 Lecture 3
Radiation of SoundThe Point Source
Unbounded Homogeneous Medium
U(t)
ra
Simple Point Source
ADVANCED STRUCTURAL DYNAMICS AND ACOUSTICS13.811 Lecture 3
Point Source in Halfspace
ADVANCED STRUCTURAL DYNAMICS AND ACOUSTICS13.811 Lecture 3
Point Source in Halfspace
50
40
30
20
10
00.2 0.4 0.6 0.8 1.0
Horizontal wavenumber (m-1)
Mag
nitu
de
F = 150 Hz
ADVANCED STRUCTURAL DYNAMICS AND ACOUSTICS13.811 Lecture 3
Point Source in Halfspace
