An acoustic analogy is an exact rearrangement of the fluid equations into a chosen linear propagation operator acting on an acoustic variable, with everything left over placed on the right as effective forcing. The rearrangement becomes a sound-prediction method only after a reference medium, boundary conditions and approximations to the forcing are specified. In particular, a right-hand-side term need not represent independently generated sound.

Use Einstein summation convention and write

\partial_{i} = \partial / \partial x_{i}

. Differentiating the continuity equation in time and taking the divergence of the momentum equation eliminates

\partial_{t} (ρ u_{i})

ρ_{tt} = \partial_{i} \partial_{j} (ρ u_{i} u_{j} - σ_{ij}) + \nabla^{2} (p + χ) .

(1)

Set

q = p - p_{0}

and

W_{ij} = ρ u_{i} u_{j} - σ_{ij}

. Since

p + χ = q + P

, the constant

P

has zero Laplacian. The prescribed reference mass density and reference speed of sound are independent of time, so

Q_{tt} = \frac{q _{tt}}{c _{0}^{2}} - ρ_{tt} .

(2)

Combining the two identities gives

(c_{0}^{- 2} \partial_{t}^{2} - \nabla^{2}) q = \partial_{i} \partial_{j} W_{ij} + \partial_{t}^{2} Q .

(3)

Only conservation of mass and conservation of momentum have been used; no equation of state or energy equation was needed. Spatial variation of

c_{0}

creates no omitted derivative in this identity, because it multiplies a time derivative. The double divergence of the momentum flux tensor has the structure of an acoustic quadrupole.

For a localized flow, choose the reference fields to match the stationary surrounding medium:

ρ_{0} = ρ_{0}

and

c_{0}^{2} = c_{0}^{2}

outside the flow, with

P

chosen so

p_{0} = p_{0}

there. A uniform surrounding fluid permits ambient constant mass density and adiabatic sound speed, giving the familiar homogeneous wave equation. A nonuniform surrounding fluid calls for its actual stationary reference profiles, extended sensibly through the flow region. This makes the acoustic variable vanish in the unperturbed exterior and minimizes artificial contrast terms. In a uniform isentropic exterior the leading acoustic relation

p^{'} = c_{0}^{2} ρ^{'}

also makes

Q

vanish to first order. In a stratified exterior, propagation and entropy-advection effects can remain in

Q

, as the next part demonstrates.

b

 0  0

Solution

 0  0

At rest, all time derivatives and advective terms vanish. The continuity equation and the stated energy equation are then identities, while the momentum equation reduces to

\nabla (p_{0} + χ) = 0

. The reference state need not have uniform mass density or entropy. Assume smooth positive reference pressure and mass density, and retain only first-order perturbations.

The linearized continuity equation, momentum balance and adiabatic energy relation are

ρ_{t}^{'} + u^{'} \cdot \nabla ρ_{0} + ρ_{0} \nabla \cdot u^{'} = 0, ρ_{0} u_{t}^{'} = - \nabla p^{'},

(1)

p_{t}^{'} + u^{'} \cdot \nabla p_{0} = c_{0}^{2} (ρ_{t}^{'} + u^{'} \cdot \nabla ρ_{0}) = - ρ_{0} c_{0}^{2} \nabla \cdot u^{'} .

(2)

The perturbation of

c^{2}

multiplies a vanishing reference material derivative, so it does not enter at first order. Differentiate the last equation in time, then substitute the linearized momentum equation:

p_{tt}^{'} = \frac{\nabla p _{0}}{ρ _{0}} \cdot \nabla p^{'} + ρ_{0} c_{0}^{2} \nabla \cdot (\frac{\nabla p ^{'}}{ρ _{0}}) .

(3)

Expanding the divergence yields the stratified acoustic pressure equation

\frac{p _{tt}^{'}}{c _{0}^{2}} - \nabla^{2} p^{'} = (\frac{\nabla p _{0}}{c _{0}^{2} ρ _{0}} - \frac{\nabla ρ _{0}}{ρ _{0}}) \cdot \nabla p^{'} .

(4)

For a perfect gas,

c_{0}^{2} ρ_{0} = γ p_{0}

. Put

a (x) = p_{0}^{1/ γ} / ρ_{0}

. Then

\nabla lo g a = \frac{\nabla p _{0}}{γ p _{0}} - \frac{\nabla ρ _{0}}{ρ _{0}}, \frac{1}{a} \nabla \cdot (a \nabla p^{'}) = \nabla^{2} p^{'} + \nabla lo g a \cdot \nabla p^{'} .

(5)

Thus the divergence-form stratified acoustic pressure equation is

c_{0}^{- 2} p_{tt}^{'} - a^{- 1} \nabla \cdot (a \nabla p^{'}) = 0.

(6)

This is homogeneous linear propagation through the static medium, including its inhomogeneity. In the acoustic analogy of part (a),

W

has no first-order contribution when viscosity is neglected and the reference velocity is zero. Nevertheless the chosen left-hand operator lacks the gradient terms in the genuine propagation equation. Those effects must consequently appear in

Q_{tt}

. They describe propagation of an existing disturbance, rather than an independent source. Since reference-field choices also change the division between the operator and forcing, $Q_{tt}$ cannot be identified unambiguously with newly generated noise. Its presence in the right-hand side is a consequence of the chosen analogy, not a source-classification theorem.