Acoustic wave equation

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


:: { 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.


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.


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)


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


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


:: 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.


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

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