# 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 orderpartial differential equation . The equation describes the evolution ofacoustic pressure "p" orparticle velocity **u**as a function of space**r**and time "t". TheSI unit of measure for pressure is the pascal, and for velocity is themeter 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

(the local deviation from the ambient pressure), and where $c$ is theacoustic pressure 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 itswave 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

*Wikimedia Foundation.
2010.*

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