Linearized Potential Flow
The equations presented here are the linearized potential flow equations. This restricts the applications of the physics interface to systems where the background flow is well described by a compressible potential flow, that is, a flow that is inviscid, barotropic, and irrotational. The sound sources also need to be external to the flow or at least they need to be represented by simple well defined sources. Application areas typically include modeling of how jet engine noise is influenced by the mean flow.
The basic dependent variable is the velocity potential conventionally defined by the relationship
where = u(rt) is the particle velocity associated with the acoustic wave motion. The total  particle velocity is given by
(4-9)
where V denotes the local mean velocity for the fluid motion (the mean flow is labeled u0 in the linearized Euler and Navier-Stokes interfaces). The dynamic equations for this mean-flow field are described in the next subsection. For now, just assume V to be a given irrotational background velocity field; hence, also the mean-flow velocity can be defined in terms of a potential field Φ, by V = ∇Φ. 
The linearized equation for the velocity potential , governing acoustic waves in a background flow with mean background velocity V, mean background density ρ0, and mean background speed of sound c0, is
(4-10)
In deriving this equation, all variables appearing in the full nonlinear fluid-dynamics equations were first split in time-independent and acoustic parts, in the manner of Equation 4-9. Then, linearizing the resulting equations in the acoustic perturbation and eliminating all acoustic variables except the velocity potential gives Equation 4-10. Thus, the density ρ in this equation is the time-independent part. The corresponding acoustic part is ρ(rt) = p(rt)/c02 where p is the acoustic pressure, given by
Hence, once Equation 4-10 has been solved for the velocity potential, the acoustic pressure can easily be calculated.
When transformed to the frequency domain, the wave Equation 4-10 reads
while the acoustic pressure is
Typical boundary conditions include:
Frequency Domain Equations
In the frequency domain the velocity potential φ is assumed to be a harmonic wave of the form
The governing frequency domain — or time-harmonic — equation is
In 2D, where
the out-of-plane wave number kz enters the equations when the  operators are expanded:
The default value of the out-of-plane wave number is 0, that is, no wave propagation perpendicular to the 2D plane. In a mode analysis the equations are solved for kz.
For 2D axisymmetric components
the azimuthal wave number m similarly  appears in the equation as a parameter:
The background velocity field V cannot have an azimuthal component because the flow is irrotational.
Time-Dependent Equation
In the time domain, the physics interface solves for the velocity potential with an arbitrary transient dependency. The following equation governs the acoustic waves in a mean potential flow:
(4-11)
Here ρ0 (SI unit: kg/m3) is the background mean flow density, V (SI unit: m/s) denotes the background mean velocity, and c0 (SI unit: m/s) refers to the speed of sound. The software solves the equation for the velocity potential , with SI unit m2/s. The validity of this equation relies on the assumption that ρ0, V, and c0 are approximately constant in time, while they can be functions of the spatial coordinates.
The background velocity field V cannot have an azimuthal component because the flow is irrotational.
Boundary Mode Analysis
The boundary mode analysis type in 3D uses the eigenvalue solver to solve the equation
(4-12)
for the eigenmodes, , and eigenvalues, λ = −ikz, on a bounded two-dimensional domain, Ω, given well-posed edge conditions on ∂Ω. In this equation, is the velocity potential, ρ0 is the background mean flow density, c0 is the speed of sound, ω is the angular frequency, and kz is the out-of-plane wave number or propagation constant. Furthermore, Vt denotes the background mean velocity in the tangential plane while Vn is the background mean velocity component in the normal direction.
Although the out-of-plane wave number is called kz, the two-dimensional surface on which Equation 4-12 is defined does not necessarily have to be normal to the  z-axis for 3D geometries.