solutions to the wave equation

what is a solution to the wave equation

any function that propagates: pressure(x,t)=f(x\pm ct)

what happens when you differentiate the solution to the wave equation

if you differentiate by time twice: pressure(x,t)=c^2f’’(x\pm ct)

if you differentiate by space the equation does not change

difference between a one way and two way wave equation

in the one way wave equation: \frac{d^{2}P}{dt^{2}}-c^{2}\frac{d^{2}P}{dx^{2}} → C² can~be~either~(+C)²~or~(-C²) the direction is not specified

therefore the solution is the sum of both left and right moving waves: pressure(x,t)=f(x+ct)+f(x-ct)

in the two way wave equation the direction of wave propagation needs to be specified:

\frac{d^{2}P}{dt^{2}}-c^2\frac{d^2P}{dx^2}=\left(\frac{\partial}{\partial t}-c\frac{\partial}{\partial x})(\frac{\partial}{\partial t}+c\frac{\partial}{\partial x})\right.p=0

how to check if something is a solution to the wave equation

1. take the 2nd order derivative of the function with respects to space

2. take the 2nd order derivative of the function with respects to time

3. substitute both derivatives into the 1D wave equation: \frac{1}{c_{0^{}}^2}\frac{d^{2}P}{dt^{2}}=\frac{d^2P}{dx^2}

   1. if both sides are equal to each other then the equation is a solution to the wave equation

what does it mean if something is a solution to the wave equation

it means that the equation is describing a possible wave pattern than moves through space and time according to the physics of wave propagation

• c term describes the speed of propagation

• +ve → moving left

• -ve → moving right

what is a single frequency harmonic wave

it is a sinusoidal wave that propagates in 1 direction without changing amplitude and has a single constant frequency

how can a harmonic wave be written

by function of space, where k is the wavenumber:

acos(kx)~~~~~~~~~asin(kx)~~~~~~~~~ae^{ikx}

by function of time, where w is the angular frequency:

acos(wt)~~~~~~~~~asin(wt)~~~~~~~~~ae^{iwt}

as a propagating wave:

acos(kx\pm wt)~~~~~~~~~asin(kx\pm wt)~~~~~~~~~ae^{ikx\pm wt}

for the exponent form:

• real component is the

• the imaginary component is the phase of the wave

propagation direction of a single frequency harmonic plane wave

(kx\pm wt) the value inside the bracket need to stay constant

therefore:

• (kx-wt): as w increases k also has to increase → moves in +ve x direction

• (kx+wt) : as w increases k has to become more negative → moves in -ve x direction

• (-kx-wt) :as w increases k has to become more negative → moves in -ve direction

• (-kx+ wt): as w increases k also increases _> moves in +ve direction

when is a single frequency harmonic wave a solution to the wave equation

if k=\pm\frac{w}{c} or if w=\pm ck

2D and 3D harmonic waves

k and x have (x,y,z) components

the wavenumber = \sqrt{k_x²+k_y²+k_z²}

the equation is now ae^{i(k\cdot x\pm wt)} → since K now has component it’s now a dot product

plane way is not a solution is w=c|k|

what is fourier or K space?

• is the x, y and z components of the wavenumber plotted onto a graph

• each point contains frequency and phase information

• decomposes any sound filed into a sum of different harmonic plane waves

\Sigma P(k_x,K_y,k_z)e^{k_xX+k_yY+k_zZ}

where:

• P(…): amplitude of each directional component

• exponent: plane wave in each dimension

what is superposition

the sum of solutions to the wave equation is itself a solution to the wave equation as the wave equation is linear

what can be used to model the solution of a very complex wave

any complex sound wave can be described as a sum of point sources (dirac delta function) but the wave function P must be the green’s function for the equation to work

what is the green’s function, how can else can it be written

it is the solution for a wave equation which has dirac delta functions as a source

G(x,t)

1D green function

\frac{c_0}{2}\delta(t-\frac{|x|}{c_0})

2D green function

\frac{\delta(t-\frac{|x|}{c_0})}{2\pi\sqrt{t²+\frac{|x|^2}{c_o²}}}

3D green function

\frac{\delta(t-\frac{|x|}{c_0})}{4\pi\left|x\right|}

why is the green’s function usefull

can be used to model a very complex source (ex: non-uniformly shaped) as a sum of point sources

how do you write out a complex source as a sum of green’s functions

what are the type of numerical solutions to the wave equation