Mathematical Models for Acoustic Analysis
Standard acoustic problems involve solving for the small acoustic pressure variations (often denoted p’ or p1) on top of the stationary/quiescent background pressure (often denoted p0 or pA). Mathematically this represents a linearization (small parameter expansion) around the stationary quiescent values.
The governing equations, for a compressible lossless (no thermal conduction and no viscosity) fluid flow problem, are the momentum equation (Euler’s equation) and the continuity equation. These are given by:
where ρ is the total density, p is the total pressure, and u is the total velocity field. In classical pressure acoustics all thermodynamic processes are assumed reversible and adiabatic, known as an isentropic process. The small parameter expansion is performed on a stationary fluid of density ρ0 (SI unit: kg/m3) and at pressure p0 (SI unit: Pa) such that:
where the subscript 1 represent the small acoustic variations (sometimes these are denoted with a prime instead). Inserting these into the governing equations and only retaining terms linear in the small perturbation variables yields
One of the dependent variables, the density, is removed by expressing it in terms of the pressure using the density differential (linearization)
where cs is recognized as the (isentropic) speed of sound (SI unit: m/s) at constant entropy s. It should be noted that this equation is valid for constant valued (not space dependent) background density ρ0 and background pressure p0. The subscripts s is dropped in the following. From the above expression it also follows that another requirement for linear acoustics (the perturbation approximation) to be valid is that
Finally, rearranging the equations (divergence of momentum equation inserted into the continuity equation) and dropping the subscript 1 yields the wave equation for sound waves in a lossless medium
(1-1)
The speed of sound is related to the compressibility of the fluid where the waves are propagating. The combination ρ  c2 is called the bulk modulus, commonly denoted K (SI unit: N /m2).
A special case is a time-harmonic wave, for which the pressure varies with time as
where ω = 2π   f (SI unit: rad/s)  is the angular frequency and  f (SI unit: Hz) is denoting the frequency. The wave equation for acoustic waves reduces to the Helmholtz equation:
(1-2)
where the ratio ω/c is recognized as the wave number k. This equation can also be treated as an eigenvalue PDE to solve for eigenmodes and eigenfrequencies.
Typical boundary conditions for the wave equation and the Helmholtz equation are:
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Sound-hard boundaries (walls)
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Sound-soft boundaries
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Impedance boundary conditions
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Radiation boundary conditions
A detailed derivation of the governing equations is given in Theory Background for the Pressure Acoustics Branch. For the propagation of compressional (acoustic) waves in a viscous and thermally conducting fluid the theory is presented in Theory Background for the Thermoviscous Acoustics Branch and for acoustics in moving media (aeroacoustics) in Theory Background for the Aeroacoustics Branch.