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 pressure p or 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 N dimensions
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

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

[edit] Solution

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

[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:

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