Acoustic wave equation

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")

Wave equation

Acoustic wave equation in one dimension

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.

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

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)

::$PV=nRT$

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

::$P\; =\; C\; ho\; ,$

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

::$P\; -\; P\_0\; =\; left(frac\{partial\; P\}\{partial\; ho\}\; ight)\; (\; ho\; -\; ho\_0)$.

The adiabatic bulk modulus for a fluid is defined as

::$B=\; ho\_0\; left(frac\{partial\; P\}\{partial\; ho\}\; ight)\_\{adiabatic\}$

which gives the result

::$P-P\_0=B\; frac\{\; ho\; -\; ho\_0\}\{\; ho\_0\}$.

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

::$s\; =\; frac\{\; ho\; -\; ho\_0\}\{\; ho\_0\}$

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

::$frac\{partial\; ho\}\{partial\; t\}\; +\; frac\{partial\; \}\{partial\; x\}\; (\; ho\; u)\; =\; 0$.

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

::$frac\{partial\}\{partial\; t\}\; (\; ho\_0\; +\; ho\_0\; s)\; +\; frac\{partial\; \}\{partial\; x\}\; (\; ho\_0\; u\; +\; ho\_0\; s\; u)\; =\; 0$

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:

::$ho\; frac\{D\; u\}\{D\; t\}\; +\; frac\{partial\; P\}\{partial\; x\}\; =\; 0$,

where $D/Dt$ 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:

::$(\; ho\_0\; +\; ho\_0\; s)left(\; frac\{partial\; \}\{partial\; t\}\; +\; u\; frac\{partial\; \}\{partial\; x\}\; ight)\; u\; +\; frac\{partial\; \}\{partial\; x\}\; (P\_0\; +\; p)\; =\; 0$.

Rearranging and neglecting small terms, the resultant equation is:

::$ho\_0frac\{partial\; u\}\{partial\; t\}\; +\; frac\{partial\; p\}\{partial\; x\}\; =\; 0$.

Taking the time derivative of the continuity equation and the spacial derivative of the force equation results in:

::$frac\{partial^2\; s\}\{partial\; t^2\}\; +\; frac\{partial^2\; u\}\{partial\; x\; partial\; t\}\; =\; 0$

::$ho\_0\; frac\{partial^2\; u\}\{partial\; x\; partial\; t\}\; +\; frac\{partial^2\; p\}\{partial\; x^2\}\; =\; 0$.

Multiplying the first by $ho\_0$, subtracting the two, and substituting the linearized equation of state,

::$-\; frac\{\; ho\_0\; \}\{B\}\; frac\{partial^2\; p\}\{partial\; t^2\}\; +\; frac\{partial^2\; p\}\{partial\; x^2\}\; =\; 0$.

The final result is

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

where $c\; =\; sqrt\{\; frac\{B\}\{\; ho\_0$.

Acoustic wave equation in Homogeneous Media

Equation

::$abla\; ^2\; p\; -\; \{1\; over\; c^2\}\; \{\; partial^2\; p\; over\; partial\; t\; ^2\; \}\; =\; 0$

Solution

::$k\; =\; omega/c$

Cartesian coordinates::: $p(r,k)=Ae^\{pm\; ikr\}$.

Cylindrical coordinates::: $p(r,k)=AH\_0^\{(1)\}(kr)\; +\; BH\_0^\{(2)\}(kr)$.

where the asymptotic approximation to the Hankel functions, when $kr\; ightarrow\; infty$, are

::$H\_0^\{(1)\}(kr)\; simeq\; sqrt\{frac\{2\}\{pi\; kre^\{i(kr-pi/4)\}$

::$H\_0^\{(2)\}(kr)\; simeq\; sqrt\{frac\{2\}\{pi\; kre^\{-i(kr-pi/4)\}$.

Spherical coordinates::: $p(r,k)=frac\{A\}\{r\}e^\{pm\; ikr\}$.

Depending on the chosen Fourier convention, one of these represents on outward travelling wave and the other an unphysical inward travelling wave.

Derivation

Acoustic wave equation in non-ideal gas flow

heterogeneity, energy loss and flow speed

* Equation

* Solution

* Derivation

Acoustic wave equation in solids

See also

* Acoustics
* Wave Equation
* Differential Equations
* Thermodynamics
* Fluid Dynamics
* Pressure
* Ideal Gas Law