Acoustic wave equation
From Wikipedia, the free encyclopedia
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)
Contents |
[edit] Wave equation
[edit] Acoustic wave equation in one dimension
[edit] Equation
- <math> { \partial^2 p \over \partial x ^2 } - {1 \over c^2} { \partial^2 p \over \partial t ^2 } = 0 </math>
[edit] Solution
<math>p=p_0 \sin(\omega t \mp kx)</math>
[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)
<math>PV=nRT</math>
In an adiabatic process pressure P as a function of density <math>\rho</math> can be linearized to
<math>P = C \rho \,</math>
where C is some constant. Breaking the pressure and density into their mean and total components and noting that <math>C=\frac{\partial P}{\partial \rho}</math>:
<math>P - P_0 = \left(\frac{\partial P}{\partial \rho}\right) (\rho - \rho_0)</math>.
The adiabatic bulk modulus for a fluid is defined as
<math>B= \rho_0 \left(\frac{\partial P}{\partial \rho}\right)_{adiabatic}</math>
which gives the result
<math>P-P_0=B \frac{\rho - \rho_0}{\rho_0}</math>.
Condensation, s, is defined as the change in density for a given ambient fluid density.
<math>s = \frac{\rho - \rho_0}{\rho_0}</math>
The linearized equation of state becomes
<math>p = B s\,</math> where p is the acoustic pressure.
The continuity equation (conservation of mass) in one dimension is
<math>\frac{\partial \rho}{\partial t} + \frac{\partial }{\partial x} (\rho u) = 0</math>.
Again the equation must be linearized and the variables split into mean and variable components.
<math>\frac{\partial}{\partial t} ( \rho_0 + \rho_0 s) + \frac{\partial }{\partial x} ((\rho_0 u + \rho_0 s u) = 0</math>
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:
<math>\frac{\partial s}{\partial t} + \frac{\partial }{\partial x} u = 0</math>
Euler's Force equation (conservation of momentum) is the last needed component. In one dimension the equation is:
<math>\rho \frac{d u}{d t} + \frac{\partial P}{\partial x} = 0</math>.
Linearizing the variables:
<math>(\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</math>.
Rearranging and neglecting small terms, the resultant equation is:
<math>\rho_0\frac{\partial u}{\partial t} + \frac{\partial p}{\partial x} = 0</math>.
Taking the time derivative of the continuity equation and the spacial derivative of the force equation results in:
<math>\frac{\partial^2 s}{\partial t^2} + \frac{\partial^2 u}{\partial x \partial t} = 0</math>
<math>\rho_0 \frac{\partial^2 u}{\partial x \partial t} + \frac{\partial^2 p}{\partial x^2} = 0</math>.
Combining and substituting the linearized equation of state,
<math>- \frac{\rho_0 }{B} \frac{\partial^2 p}{\partial t^2} + \frac{\partial^2 p}{\partial x^2} = 0</math>.
The final result is
<math> { \partial^2 p \over \partial x ^2 } - {1 \over c^2} { \partial^2 p \over \partial t ^2 } = 0 </math>
where <math>c = \sqrt{ \frac{B}{\rho_0 }}</math>.
[edit] Acoustic wave equation in N dimensions
[edit] Equation
- <math> \nabla ^2 p - {1 \over c^2} { \partial^2 p \over \partial t ^2 } = 0 </math>
[edit] Solution
[edit] Derivation
[edit] Acoustic wave equation in non-ideal gas flow
heterogeneity, energy loss and flow speed
- Equation
- Solution
- Derivation
