Acoustic wave equation

The wave equation describing sound in one dimension (position ${\displaystyle x}$) is

${\displaystyle {\partial ^{2}p \over \partial x^{2}}-{1 \over c^{2}}{\partial ^{2}p \over \partial t^{2}}=0,}$

where ${\displaystyle p}$ is the acoustic pressure (the local deviation from the ambient pressure), and where ${\displaystyle c}$ is the speed of sound.[2]

Provided that the speed ${\displaystyle c}$ is a constant, not dependent on frequency (the dispersionless case), then the most general solution is

${\displaystyle p=f(ct-x)+g(ct+x)}$

where ${\displaystyle f}$ and ${\displaystyle g}$ are any two twice-differentiable functions. This may be pictured as the superposition of two waveforms of arbitrary profile, one (${\displaystyle f}$) travelling up the x-axis and the other (${\displaystyle g}$) down the x-axis at the speed ${\displaystyle c}$. The particular case of a sinusoidal wave travelling in one direction is obtained by choosing either ${\displaystyle f}$ or ${\displaystyle g}$ to be a sinusoid, and the other to be zero, giving

${\displaystyle p=p_{0}\sin(\omega t\mp kx)}$.

where ${\displaystyle \omega }$ is the angular frequency of the wave and ${\displaystyle k}$ is its wave number.

Derivation of the acoustic wave equation

The derivation of the wave equation involves three steps: derivation of the equation of state, the linearized one-dimensional continuity equation, and the linearized one-dimensional force equation.

The equation of state (ideal gas law)

${\displaystyle PV=nRT}$

In an adiabatic process, pressure P as a function of density ${\displaystyle \rho }$ can be linearized to

${\displaystyle P=C\rho \,}$

where C is some constant. Breaking the pressure and density into their mean and total components and noting that ${\displaystyle C={\frac {\partial P}{\partial \rho }}}$:

${\displaystyle P-P_{0}=\left({\frac {\partial P}{\partial \rho }}\right)(\rho -\rho _{0})}$.

The adiabatic bulk modulus for a fluid is defined as

${\displaystyle B=\rho _{0}\left({\frac {\partial P}{\partial \rho }}\right)_{adiabatic}}$

which gives the result

${\displaystyle P-P_{0}=B{\frac {\rho -\rho _{0}}{\rho _{0}}}}$.

Condensation, s, is defined as the change in density for a given ambient fluid density.

${\displaystyle s={\frac {\rho -\rho _{0}}{\rho _{0}}}}$

The linearized equation of state becomes

${\displaystyle p=Bs\,}$ where p is the acoustic pressure (${\displaystyle P-P_{0}}$).

The continuity equation (conservation of mass) in one dimension is

${\displaystyle {\frac {\partial \rho }{\partial t}}+{\frac {\partial }{\partial x}}(\rho u)=0}$.

Where u is the flow velocity of the fluid. Again the equation must be linearized and the variables split into mean and variable components.

${\displaystyle {\frac {\partial }{\partial t}}(\rho _{0}+\rho _{0}s)+{\frac {\partial }{\partial x}}(\rho _{0}u+\rho _{0}su)=0}$

Rearranging and noting that ambient density changes with neither time nor position and that the condensation multiplied by the velocity is a very small number:

${\displaystyle {\frac {\partial s}{\partial t}}+{\frac {\partial }{\partial x}}u=0}$

Euler's Force equation (conservation of momentum) is the last needed component. In one dimension the equation is:

${\displaystyle \rho {\frac {Du}{Dt}}+{\frac {\partial P}{\partial x}}=0}$,

where ${\displaystyle D/Dt}$ represents the convective, substantial or material derivative, which is the derivative at a point moving along with the medium rather than at a fixed point.

Linearizing the variables:

${\displaystyle (\rho _{0}+\rho _{0}s)\left({\frac {\partial }{\partial t}}+u{\frac {\partial }{\partial x}}\right)u+{\frac {\partial }{\partial x}}(P_{0}+p)=0}$.

Rearranging and neglecting small terms, the resultant equation becomes the linearized one-dimensional Euler Equation:

${\displaystyle \rho _{0}{\frac {\partial u}{\partial t}}+{\frac {\partial p}{\partial x}}=0}$.

Taking the time derivative of the continuity equation and the spatial derivative of the force equation results in:

${\displaystyle {\frac {\partial ^{2}s}{\partial t^{2}}}+{\frac {\partial ^{2}u}{\partial x\partial t}}=0}$

${\displaystyle \rho _{0}{\frac {\partial ^{2}u}{\partial x\partial t}}+{\frac {\partial ^{2}p}{\partial x^{2}}}=0}$.

Multiplying the first by ${\displaystyle \rho _{0}}$, subtracting the two, and substituting the linearized equation of state,

${\displaystyle -{\frac {\rho _{0}}{B}}{\frac {\partial ^{2}p}{\partial t^{2}}}+{\frac {\partial ^{2}p}{\partial x^{2}}}=0}$.

The final result is

${\displaystyle {\partial ^{2}p \over \partial x^{2}}-{1 \over c^{2}}{\partial ^{2}p \over \partial t^{2}}=0}$

where ${\displaystyle c={\sqrt {\frac {B}{\rho _{0}}}}}$ is the speed of propagation.

Feynman[3] provides a derivation of the wave equation for sound in three dimensions as

${\displaystyle \nabla ^{2}p-{1 \over c^{2}}{\partial ^{2}p \over \partial t^{2}}=0,}$

where ${\displaystyle \nabla ^{2}}$ is the Laplace operator, ${\displaystyle p}$ is the acoustic pressure (the local deviation from the ambient pressure), and ${\displaystyle c}$ is the speed of sound.

A similar looking wave equation but for the vector field particle velocity is given by

${\displaystyle \nabla ^{2}\mathbf {u} \;-{1 \over c^{2}}{\partial ^{2}\mathbf {u} \; \over \partial t^{2}}=0}$.

In some situations, it is more convenient to solve the wave equation for an abstract scalar field velocity potential which has the form

${\displaystyle \nabla ^{2}\Phi -{1 \over c^{2}}{\partial ^{2}\Phi \over \partial t^{2}}=0}$

and then derive the physical quantities particle velocity and acoustic pressure by the equations (or definition, in the case of particle velocity):

${\displaystyle \mathbf {u} =\nabla \Phi \;}$,

${\displaystyle p=-\rho {\partial \over \partial t}\Phi }$.

The following solutions are obtained by separation of variables in different coordinate systems. They are phasor solutions, that is they have an implicit time-dependence factor of ${\displaystyle e^{i\omega t}}$ where ${\displaystyle \omega =2\pi f}$ is the angular frequency. The explicit time dependence is given by

${\displaystyle p(r,t,k)=\operatorname {Real} \left[p(r,k)e^{i\omega t}\right]}$

Here ${\displaystyle k=\omega /c\ }$ is the wave number.

${\displaystyle p(r,k)=Ae^{\pm ikr}}$.

${\displaystyle p(r,k)=AH_{0}^{(1)}(kr)+\ BH_{0}^{(2)}(kr)}$.

where the asymptotic approximations to the Hankel functions, when ${\displaystyle kr\rightarrow \infty }$, are

${\displaystyle H_{0}^{(1)}(kr)\simeq {\sqrt {\frac {2}{\pi kr}}}e^{i(kr-\pi /4)}}$

${\displaystyle H_{0}^{(2)}(kr)\simeq {\sqrt {\frac {2}{\pi kr}}}e^{-i(kr-\pi /4)}}$.

${\displaystyle p(r,k)={\frac {A}{r}}e^{\pm ikr}}$.

Depending on the chosen Fourier convention, one of these represents an outward travelling wave and the other a nonphysical inward travelling wave. The inward travelling solution wave is only nonphysical because of the singularity that occurs at r=0; inward travelling waves do exist.