Engineering Acoustics/Acoustic wave equation

To derive the wave equation, three relations are combined: The equation of state (ideal gas law), continuity equation (conservation of mass) , and Newton's law (conservation of momentum). From the speed of sound, a relation between the acoustic pressure and the bulk modulus can be derived. The bulk modulus makes implicit use of the ideal gas law. The definition of the condensation, relative change of density in a fluid, denoted by s is also introduced.

${\displaystyle C_{o}^{2}={\frac {\partial P}{\partial \rho }}={\frac {p}{\varrho }}={\frac {p}{\rho -\rho _{o}}}}$

${\displaystyle {\frac {p}{\rho _{o}}}=C_{o}^{2}{\Big (}{\frac {\rho -\rho _{o}}{\rho _{o}}}{\Big )}}$

${\displaystyle p=\rho _{o}C_{o}^{2}s=\gamma P_{o}s=Bs}$

The general form of the continuity equation for a control volume from fluid dynamics is simplified to its one dimensional form in Cartesian coordinates.

${\displaystyle {\frac {\partial \rho }{\partial t}}+\mathbf {\nabla } (\rho \mathbf {U} )=0}$

${\displaystyle {\frac {\partial \rho }{\partial t}}+\rho _{o}{\frac {\partial U}{\partial x}}=0}$

${\displaystyle {\frac {\partial U}{\partial x}}={\frac {-1}{\rho _{o}}}{\frac {\partial \rho }{\partial t}}={\frac {-1}{\rho _{o}C_{o}^{2}}}{\frac {\partial P}{\partial t}}}$

Pressure on fluid element inside a tube

Using Newton's law on the fluid element, the net force acting on the fluid boundaries causes an acceleration on the fluid proportional to its mass.

${\displaystyle A(P(x)-P(x+dx))=(Adx\rho _{o}){\frac {dU}{dt}}}$

as ${\displaystyle dx}$ approaches zero,

${\displaystyle {\frac {P(x)-P(x+dx)}{dx}}={\frac {\partial P}{\partial x}}}$

Evaluating the derivative and neglecting small terms,

${\displaystyle {\frac {\partial P}{\partial x}}dx=(\rho _{o}dx){\frac {dU}{dt}}=(\rho _{o}dx){\Big (}{\frac {\partial U}{\partial t}}+{\frac {\partial U}{\partial x}}{\frac {dx}{dt}}{\Big )}}$

${\displaystyle {\frac {\partial U}{\partial t}}={\frac {-1}{\rho _{o}}}{\frac {\partial P}{\partial x}}}$

To obtain the wave equation, the partial derivative with respect to time is taken for the continuity equation and with respect to space for the conservation of momentum equation. The results are then equated.

${\displaystyle {\frac {\partial ^{2}U}{\partial t\partial x}}={\frac {-1}{\rho _{o}C_{o}^{2}}}{\frac {\partial ^{2}P}{\partial t^{2}}}={\frac {-1}{\rho _{o}}}{\frac {\partial ^{2}P}{\partial x^{2}}}}$

${\displaystyle {\frac {\partial ^{2}P}{\partial x^{2}}}={\frac {1}{C_{o}^{2}}}{\frac {\partial ^{2}P}{\partial t^{2}}}}$

The equation above is the acoustic wave equation in its one-dimensional form. It can be generalized to 3-D Cartesian coordinates.${\displaystyle {\frac {\partial ^{2}P}{\partial x^{2}}}+{\frac {\partial ^{2}P}{\partial y^{2}}}+{\frac {\partial ^{2}P}{\partial z^{2}}}-{\frac {1}{C_{o}^{2}}}{\frac {\partial ^{2}P}{\partial t^{2}}}=0}$

Using the Laplace operator, it can be generalized to other coordinate systems.

${\displaystyle \nabla ^{2}P-{\frac {1}{C_{o}^{2}}}{\frac {\partial ^{2}P}{\partial t^{2}}}=0}$

Original document[1]

1. [1]