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# 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

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

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

::$p = f\left(x - c t\right) + g\left(x + c t\right)$

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

::$p=p_0 sin\left(omega t mp kx\right)$.

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 $C=frac\left\{partial P\right\}\left\{partial ho\right\}$:

::$P - P_0 = left\left(frac\left\{partial P\right\}\left\{partial ho\right\} ight\right) \left( ho - ho_0\right)$.

The adiabatic bulk modulus for a fluid is defined as

::$B= ho_0 left\left(frac\left\{partial P\right\}\left\{partial ho\right\} ight\right)_\left\{adiabatic\right\}$

which gives the result

::$P-P_0=B frac\left\{ ho - ho_0\right\}\left\{ ho_0\right\}$.

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

::$s = frac\left\{ ho - ho_0\right\}\left\{ ho_0\right\}$

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

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

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

::$frac\left\{partial\right\}\left\{partial t\right\} \left( ho_0 + ho_0 s\right) + frac\left\{partial \right\}\left\{partial x\right\} \left( ho_0 u + ho_0 s u\right) = 0$

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:

::$frac\left\{partial s\right\}\left\{partial t\right\} + frac\left\{partial \right\}\left\{partial x\right\} u = 0$

Euler's Force equation (conservation of momentum) is the last needed component. In one dimension the equation is:

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

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:

::$\left( ho_0 + ho_0 s\right)left\left( frac\left\{partial \right\}\left\{partial t\right\} + u frac\left\{partial \right\}\left\{partial x\right\} ight\right) u + frac\left\{partial \right\}\left\{partial x\right\} \left(P_0 + p\right) = 0$.

Rearranging and neglecting small terms, the resultant equation is:

::$ho_0frac\left\{partial u\right\}\left\{partial t\right\} + frac\left\{partial p\right\}\left\{partial x\right\} = 0$.

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

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

::$ho_0 frac\left\{partial^2 u\right\}\left\{partial x partial t\right\} + frac\left\{partial^2 p\right\}\left\{partial x^2\right\} = 0$.

Multiplying the first by $ho_0$, subtracting the two, and substituting the linearized equation of state,

::$- frac\left\{ ho_0 \right\}\left\{B\right\} frac\left\{partial^2 p\right\}\left\{partial t^2\right\} + frac\left\{partial^2 p\right\}\left\{partial x^2\right\} = 0$.

The final result is

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

where $c = sqrt\left\{ frac\left\{B\right\}\left\{ ho_0$.

Acoustic wave equation in Homogeneous Media

Equation

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

Solution

::$k = omega/c$

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

Cylindrical coordinates::: $p\left(r,k\right)=AH_0^\left\{\left(1\right)\right\}\left(kr\right) + BH_0^\left\{\left(2\right)\right\}\left(kr\right)$.

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

::$H_0^\left\{\left(1\right)\right\}\left(kr\right) simeq sqrt\left\{frac\left\{2\right\}\left\{pi kre^\left\{i\left(kr-pi/4\right)\right\}$

::$H_0^\left\{\left(2\right)\right\}\left(kr\right) simeq sqrt\left\{frac\left\{2\right\}\left\{pi kre^\left\{-i\left(kr-pi/4\right)\right\}$.

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

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

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

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