Plane Waves and Geometric Spreading Flashcards

1
Q

What is the form on the solutions to the 1D wave equation & plane acoustic pressure waves?

A

p(x,t) = f(x ± ct)

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2
Q

Which direction does f(x - ct) go?

A

in positive x - direction

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3
Q

How are one-way wave equations found?

A

by equating the 1D wave equation to zero and separating the dirac derivatives

with solutions f(x ± ct)

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4
Q

Which direction does f(x + ct) go?

A

in negative x - direction

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5
Q

How can functions that vary sinusoidally in space be written?

A

a sin(kx)
a cos(kx)
Re (a e^ikx)

a = amplitude
k = wave vector

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6
Q

How can functions that vary sinusoidally in time be written?

A

a sin(wt)
a cos(wt)
Re (a e^iwt)

w = temporal frequency

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7
Q

What is a single frequency harmonic wave and what are its solutions?

A

A propagating sinusoid

a sin(kx - wt)
a cos(kx - wt)
a e^i(kx - wt)

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8
Q

What does a negative k correspond to?

A

Propagation in negative x-direction

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9
Q

What does a positive k correspond to?

A

Propagation in positive x-direction

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10
Q

When is an equation a solution to the wave equation?

A

when k = ±w/c

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11
Q

What is the wave vector in 1D?

A

k or - k

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12
Q

What gives the direction of a plane wave?

A

wave vector

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13
Q

How can an acoustic field (e.g image) be constructed?

A

As a weighted sum of plane waves

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14
Q

What is superposition?

A

A term used to mean that a sum of solutions to the wave equation is itself a solution (because wave equation is linear)

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15
Q

What is the characteristic acoustic impedence?

A

Z = ρ_0 c_0

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16
Q

How does energy spread out over an expanding spherical wavefront?

A

The intensity follows an inverse square law

17
Q

What is the from of spherical acoustic pressure waves?

A

g(r ± ct) /r

18
Q

What is a line source?

A

A continuous line of point sources in 3D

19
Q

What are cylindrical waves?

A

Waves emanating from a line source (energy is spread out over the expanding cylindrical wavefront)

20
Q

What is the equivalent of a line source in 3D?

A

A point source in 2D

21
Q

What is the form of the cylindrical acoustic pressure wave?

A

g(r ± ct)/ √r

22
Q

How is a plane wave generated and what form do plane acoustic pressure waves have?

A

A continuous plane of point sources in 3D

g(r ± ct)

23
Q

How does acoustic energy decay with distance?

A

Due to geometric spreading, absorption and scattering

24
Q

What is intensity magnitude P/S (W/m^2) proportional to?

25
How do plane waves change over surface area, intensity and pressure?
constant no decay no decay
26
How do cylindrical waves change over surface area, intensity and pressure?
2π rh 1/r 1/ √r
27
How do spherical waves change over surface area, intensity and pressure?
4π r^2 1/r^2 1/r