Acoustic wave equation

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In physics, the acoustic wave equation governs the propagation of acoustic waves through a material medium. The form of the equation is a second order partial differential equation. The equation describes the evolution of acoustic pressure p or particle velocity u as a function of space r and time t. The SI unit of measure for pressure is the pascal, and for velocity is the meter per second.

A simplified form of the equation describes acoustic waves in only one spatial dimension (position x), while a more sophisticated form describes waves in three dimensions (displacement vector r = (x,y,z)).

p = p(r,t) = p(x,y,z,t)

AND

u = u(r,t) = u(x,y,z,t)

1 Wave equation
2 Acoustic wave equation in one dimension
3 Acoustic wave equation in Homogeneous Media
4 Acoustic wave equation in non-ideal gas flow
5 Acoustic wave equation in solids
6 See also

[edit] Wave equation

[edit] Acoustic wave equation in one dimension

[edit] Equation

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

where $p$ is the acoustic pressure (the local deviation from the ambient pressure), and where $c$ is the speed of sound.

[edit] Solution

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

p = f (x - c t) + g (x + c t)

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

$p=p_0 \sin(\omega t \mp kx)$ .

where $ω$ is the angular frequency of the wave and $k$ is its wave number.

[edit] Derivation

The wave equation can be developed from the linearized one-dimensional continuity equation, the linearized one-dimensional force equation and the equation of state.

The equation of state (ideal gas law)

P V = n R T

In an adiabatic process, pressure P as a function of density $ρ$ can be linearized to

$P = C \rho \,$

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

$P - P_0 = \left(\frac{\partial P}{\partial \rho}\right) (\rho - \rho_0)$ .

The adiabatic bulk modulus for a fluid is defined as

$B= \rho_0 \left(\frac{\partial P}{\partial \rho}\right)_{adiabatic}$

which gives the result

$P-P_0=B \frac{\rho - \rho_0}{\rho_0}$ .

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

$s = \frac{\rho - \rho_0}{\rho_0}$

The linearized equation of state becomes

$p = B s\,$ where p is the acoustic pressure.

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

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

Again the equation must be linearized and the variables split into mean and variable components.

$\frac{\partial}{\partial t} ( \rho_0 + \rho_0 s) + \frac{\partial }{\partial x} (\rho_0 u + \rho_0 s u) = 0$

Rearranging and noting that ambient density does not change with time or position and that the condensation multiplied by the velocity is a very small number:

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

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

$\rho \frac{D u}{D t} + \frac{\partial P}{\partial x} = 0$ ,

where $D / D t$ represents the convective, substantial or material derivative, which is the derivative at a point moving with medium rather than at a fixed point.

Linearizing the variables:

$(\rho_0 +\rho_0 s)\left( \frac{\partial }{\partial t} + u \frac{\partial }{\partial x} \right) u + \frac{\partial }{\partial x} (P_0 + p) = 0$ .