Plane Waves and Geometric Spreading Flashcards
What is the form on the solutions to the 1D wave equation & plane acoustic pressure waves?
p(x,t) = f(x ± ct)
Which direction does f(x - ct) go?
in positive x - direction
How are one-way wave equations found?
by equating the 1D wave equation to zero and separating the dirac derivatives
with solutions f(x ± ct)
Which direction does f(x + ct) go?
in negative x - direction
How can functions that vary sinusoidally in space be written?
a sin(kx)
a cos(kx)
Re (a e^ikx)
a = amplitude
k = wave vector
How can functions that vary sinusoidally in time be written?
a sin(wt)
a cos(wt)
Re (a e^iwt)
w = temporal frequency
What is a single frequency harmonic wave and what are its solutions?
A propagating sinusoid
a sin(kx - wt)
a cos(kx - wt)
a e^i(kx - wt)
What does a negative k correspond to?
Propagation in negative x-direction
What does a positive k correspond to?
Propagation in positive x-direction
When is an equation a solution to the wave equation?
when k = ±w/c
What is the wave vector in 1D?
k or - k
What gives the direction of a plane wave?
wave vector
How can an acoustic field (e.g image) be constructed?
As a weighted sum of plane waves
What is superposition?
A term used to mean that a sum of solutions to the wave equation is itself a solution (because wave equation is linear)
What is the characteristic acoustic impedence?
Z = ρ_0 c_0
How does energy spread out over an expanding spherical wavefront?
The intensity follows an inverse square law
What is the from of spherical acoustic pressure waves?
g(r ± ct) /r
What is a line source?
A continuous line of point sources in 3D
What are cylindrical waves?
Waves emanating from a line source (energy is spread out over the expanding cylindrical wavefront)
What is the equivalent of a line source in 3D?
A point source in 2D
What is the form of the cylindrical acoustic pressure wave?
g(r ± ct)/ √r
How is a plane wave generated and what form do plane acoustic pressure waves have?
A continuous plane of point sources in 3D
g(r ± ct)
How does acoustic energy decay with distance?
Due to geometric spreading, absorption and scattering
What is intensity magnitude P/S (W/m^2) proportional to?
to p^2
How do plane waves change over surface area, intensity and pressure?
constant
no decay
no decay
How do cylindrical waves change over surface area, intensity and pressure?
2π rh
1/r
1/ √r
How do spherical waves change over surface area, intensity and pressure?
4π r^2
1/r^2
1/r
