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


Wikimedia Foundation. 2010.

Look at other dictionaries:

  • Wave equation — Not to be confused with Wave function. The wave equation is an important second order linear partial differential equation for the description of waves – as they occur in physics – such as sound waves, light waves and water waves. It arises in… …   Wikipedia

  • Acoustic theory — is the field relating to mathematical description of sound waves. It is derived from fluid dynamics. See acoustics for the engineering approach.The propagation of sound waves in a fluid (such as air) can be modeled by an equation of motion… …   Wikipedia

  • Acoustic analogy — Acoustic analogies are applied mostly in numerical aeroacoustics to reduce aeroacoustic sound sources to simple emitter types. They are therefore often also referred to as aeroacoustic analogies.In general, aeroacoustic analogies are derived from …   Wikipedia

  • Wave — A wave is a disturbance that propagates through space and time, usually with transference of energy. While a mechanical wave exists in a medium (which on deformation is capable of producing elastic restoring forces), waves of electromagnetic… …   Wikipedia

  • Acoustic resonance — is the tendency of an acoustic system to absorb more energy when the frequency of its oscillations matches the system s natural frequency of vibration (its resonance frequency ) than it does at other frequencies. A resonant object will probably… …   Wikipedia

  • Acoustic transmission lines — An acoustic transmission line is the acoustic analog of the electrical transmission line, typically thought of as a rigid walled tube that is long and thin relative to the wavelength of sound present in it. Pipe organs, woodwinds, and the like… …   Wikipedia

  • Cnoidal wave — US Army bombers flying over near periodic swell in shallow water, close to the Panama coast (1933). The sharp crests and very flat troughs are characteristic for cnoidal waves. In fluid dynamics, a cnoidal wave is a nonlinear and exact periodic… …   Wikipedia

  • Rogue wave — This article is about the deep ocean rogue waves which occur far out at sea. For tsunami and tidal wave phenomena, see those respective articles. For other uses, see Rogue wave (disambiguation). The Draupner wave, a single giant wave measured on… …   Wikipedia

  • Plane wave — [ real part of a plane wave travelling up.] In the physics of wave propagation (especially electromagnetic waves), a plane wave (also spelled planewave) is a constant frequency wave whose wavefronts (surfaces of constant phase) are infinite… …   Wikipedia

  • Evanescent wave — Schematic representation of evanescent waves propagating along a metal dielectric interface. The charge density oscillations, when associated with electromagnetic fields, are called surface plasmon polariton waves. The exponential dependence of… …   Wikipedia


Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”

We are using cookies for the best presentation of our site. Continuing to use this site, you agree with this.