Acoustic wave equation
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 velocityu as a function of space r and time "t". The SIunit 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")
Acoustic wave equation in one dimension
where is the
acoustic pressure(the local deviation from the ambient pressure), and where is the speed of sound.
Provided that the speed is a constant, not dependent on frequency (the dispersionless case), then the most general solution is
where and are any two twice-differentiable functions. This may be pictured as the superposition of two waveforms of arbitrary profile, one () travelling up the x-axis and the other () down the x-axis at the speed . The particular case of a sinusoidal wave travelling in one direction is obtained by choosing either or to be a sinusoid, and the other to be zero, giving
where is the
angular frequencyof the wave and 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 can be linearized to
where "C" is some constant. Breaking the pressure and density into their mean and total components and noting that :
bulk modulusfor a fluid is defined as
which gives the result
Condensation, "s", is defined as the change in density for a given ambient fluid density.
The linearized equation of state becomes
:: where "p" is the acoustic pressure.
The continuity equation (conservation of mass) in one dimension is
Again the equation must be linearized and the variables split into mean and variable components.
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:
Euler's Force equation (conservation of momentum) is the last needed component. In one dimension the equation is:
where 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:
Rearranging and neglecting small terms, the resultant equation is:
Taking the time derivative of the continuity equation and the spacial derivative of the force equation results in:
Multiplying the first by , subtracting the two, and substituting the linearized equation of state,
The final result is
Acoustic wave equation in Homogeneous Media
Cartesian coordinates::: .
Cylindrical coordinates::: .
where the asymptotic approximation to the Hankel functions, when , are
Spherical coordinates::: .
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
Acoustic wave equation in solids
Ideal Gas Law
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