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

planewave is now a solution if $w=c|k|$

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 field 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

• describes how a heterogenous medium affects the echo of the incident wave as it travels from point source at x’ to receiver at x

$$G(x,t)$$

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 why is this function implicit

how is the numerical solution to the wave equation found

using the finite difference method:

1. a grid of points is overlayed on the image

2. $\frac{d}{dt}~and~ \nabla²$ in the wave equation can eb replaced by the gradient between point

what is the gradient between grid points called

finite difference approximations

what are the types of finite difference approximations

1. fowards difference $\frac{f_{i+1}-f_{i}}{\Delta x}$

2. backwards difference $\frac{f_{i}-f_{i-1}}{\Delta x}$

3. central difference $\frac{f_{i+1}-f_{i-1}}{2\Delta x}$

numerical solution using the forwards difference

if we know a few grid points $f(x+\Delta x)$

take the taylor expansion around x: $f’(x)=\frac{f\left(x+\Delta x\right)-f\left(x\right)}{\Delta x}+O\left(\Delta x\right)$

what is the big O notation

it is the error caused by truncation and the 2nd order derivative term → error in a first order accurate approximation

we truncate the expansion to the 3rd term and rearrange for the 1st term

• any expansion above the 3rd term is so small it is insignificant

error decreases linearly if we include more terms

what is the second order accurate central difference scheme

$$f’’(x)=\frac{f\left(x+\Delta x\right)+f\left(x-\Delta x\right)-2f\left(x\right)}{\left(\Delta x\right)^2}+O\left(\Delta x^2\right)$$

how does the big O change with truncation number

$$O(\Delta x^n)$$

how do you solve the wave equation with numerical solution

1. replace the 1st and 2nd order derivatives with the 2nd order finite difference equations

2. replace the function with $P_j^n$

$$P_{j}^{n+1}=2P_{j^{}}^{n}-P_{j}^{n-1}+\left(\frac{c_{o}\Delta t}{\Delta x}\right)^2\left(P_{j+1}^{n}+P_{j-1}^{n}+P_{j}^{n}\right)$$

this tells us each pressure value on the field varies with:

1. adjacents points in space and time

when does finite difference scheme breakdown

near the edge of the grid → edge values = 0

what happens if we take smaller and smaller differences in space and time between points

solution converges as differences approaches 0 → converges to the derivative (true value) then the value no longer changes

what is consistency

the limit of the finite difference scheme equation (equation approximating the gradient) gives us the actual gradient as the limit approaches 0

what is numerical dispersion

numerical error in the FDS which introduces a frequency dependent sound speed

$\frac{d^2p}{d t^2}=c_{o}\left(k\right)^2\frac{d^2p}{d x^2}$ where $c(k)=\frac{\sin c\left(\frac{k\Delta x}{2}\right)}{\sin c\left(\frac{c_{o}k\Delta t}{2}\right)}c_{o}$

→ wave changes shape due to numerical dispersion → different frequency waves travel at different speeds causing the sound wave to disperse

how is pressure information stores in FDS

1. each value of p in each area is stored as a vector producing an array at time t

2. at each time the array is updated → time stepping scheme

what is stability in time stepping scheme

restriction on the time step size

explain stability

1. your grid of points is separated by $\Delta x$

2. wave moves with distance $c_0\Delta t$ → if $\Delta t$ (timestep) is very large → sound wave moves a very large distance every loop → maths blows up

what are the stability limits

\Delta t < \frac{\Delta x}{c_0}

courant-friedrich-lewy number

CFL=\frac{c_{o}\Delta t}{\Delta x}<1 for 1D

higher dimensions CFL <\frac{1}{\sqrt{D}} where D is the number of dimensions

when is the numerical method used

1. material properties are not uniform

2. wave propagation is non-linear

3. acoustic absorption follows a frequency power law

what is peter lax’ equivalence theorem

1. consistency: numerical solution reduces to the wave equation as the limit approaches 0

2. stability: error in the numerical solution doesn’t grow without bound

3. convergence: as limit approaches 0 the FDS reaches the derivative (true solution)

if a numerical scheme is consistent and stable it always converges