The wave equation in an acoustic medium

Let x_i, where i=1,2,3 be three orthogonal directions and $\vec x$the position vector in a coordinate system associated with the three directions. Let us define $\rho = \rho \left( {\vec x} \right)$ as the mass per volume unit in the acoustic medium, $\vec u = \vec u\left( {\vec x} \right)$ as the velocity of the acoustic medium and K as the bulk modulus of the acoustic medium. The second law of dynamics states that mass $\times$ acceleration = force = - pressure gradient:

 

$$\rho \frac{{\partial u_i }}{{\partial t}} = - \frac{{\partial P}}{{\partial x_i }}.$$ (15)

Energy can be stored by compression and volume variation. If

 

$$\vec u\left( {\vec x + \delta \vec x} \right) \ne \vec u\left( {\vec x} \right),$$ (16)

we say that the flow diverges (the volume changes). This leads to a pressure variation, proportional to the divergence of the velocity:

 

$$- \frac{{\partial P}}{{\partial t}} = K\left( {\frac{{\part... ...ial x_2 }} + \frac{{\partial u_3 }}{{\partial x_3 }}} \right).$$ (17)

The wave equation in an acoustic medium can be deduced from (15) and (17) as follows. Derivate (17) with respect to time:

 

$$\frac{{\partial ^2 P}}{{\partial t^2 }} = - K\sum\limits_{i = 1}^3 {\frac{{\partial ^2 u_i }}{{\partial t\partial x_i }}}.$$ (18)

Divide (15) by $\rho$ and derivate it with respect to the axis x_i:

 

$$\frac{{\partial ^2 u_i }}{{\partial t\partial x_i }} = - \fr... ...tial x_i }}\frac{1}{\rho }\frac{{\partial P}}{{\partial x_i }}.$$ (19)

Plug (19) in (18):

 

$$\frac{{\partial ^2 P}}{{\partial t^2 }} = K\sum\limits_{i = ... ...al x_i }}\frac{1}{\rho }\frac{{\partial P}}{{\partial x_i }}}.$$ (20)

Approximation: $\rho$ is a constant that does not depend on the position vector. By denoting the acoustic waves propagation velocity through the medium by v, where

 

$$v^2 = \frac{K}{\rho },$$ (21)

we obtain the acoustic wave equation:

 

$$\frac{{\partial ^2 P}}{{\partial t^2 }} = v^2 \sum\limits_{i = 1}^3 {\frac{{\partial ^2 P}}{{\partial x_i ^2 }}}.$$ (22)