Lighthill acoustic analogy ($({\partial }_t^2-c_0^2{\nabla }^2){\rho }^{\prime }={\partial }_i{\partial }_j{T}_{ij}$)

The Lighthill acoustic analogy rewrites fluid mass and momentum conservation as a constant-speed wave equation driven by a nonlinear Lighthill stress tensor. It is an exact rearrangement before making source approximations. The outgoing field can be constructed with a retarded acoustic Green function.

If the continuity equation has mass source $M$ and the conservative momentum equation has momentum source $S_i$, elimination of momentum gives the displayed wave equation. With no imposed momentum source, its extra term is ${\partial }_{t}M$. A compact source of total mass rate $\mathcal{M}=\int MdV$ radiates the acoustic monopole ${\rho }_M^{\prime }=\dot{\mathcal{M}}(t-r/c_0)/(4\pi c_0^{2}r)$. The momentum source supplies an additional acoustic dipole; specifying mass injection alone does not fix that dipole.

For compact Lighthill stress tensor and applied-force distributions, define $S_{ij}=\int T_{ij}{d}^{3}y$ and $F_i=\int f_i{d}^{3}y$. Differentiating the retarded acoustic Green function at far distance gives the displayed leading radiation field, evaluated at $t-r/c_0$. Source compactness requires its size to be small compared with the acoustic wavelength, not merely with the observer distance.

Mass and momentum flux defects at a moving interface add surface acoustic monopole and acoustic dipole terms to Lighthill acoustic analogy. For a physical conservative fluid shock wave, the Rankine-Hugoniot conditions set both defects to zero; the shock still affects distributional derivatives of the Lighthill stress tensor. A discontinuity alone is not independent mass or force injection.

The Lighthill acoustic analogy for a moving body has a volume acoustic quadrupole, a surface acoustic monopole from acoustic thickness noise, and a surface acoustic dipole from acoustic loading noise. For an impermeable moving surface, with outward normal $n$, normal surface speed $v_n$, and fluid and body normal velocities equal, the surface mass coefficient is ${\rho }_0{v}_n$ and the surface loading is the force exerted on the fluid. Schematically, the density-source equation isThe density perturbation is understood with the chosen interior extension. Permeable-surface versions have additional mass and momentum flux terms. Source approximations must distinguish local thickness radiation from cancellation of its compact net-volume contribution.

For a fixed impermeable object, the Ffowcs Williams-Hawkings equation has a volume acoustic quadrupole and a surface acoustic dipole, with no mass-injection term. If $F$ is the total force exerted on the fluid and $Q=\int TdV$ the integrated Lighthill stress tensor, the acoustic compact-source approximation in the acoustic far field gives the displayed pressure at retarded time. A force convention directed onto the body reverses the loading sign. The retarded acoustic Green function carries $1/c_0^2$ when the wave operator is ${\partial }_t^2-c_0^2\Delta$.

Loading noise is the acoustic dipole radiation of the force exerted by a body on its surrounding fluid. For a compact, slowly moving source, the radiating acoustic density perturbation is $n\cdot \dot{\mathcal{F}}(t-R/c_0)/(4\pi c_0^{3}R)$, where $\mathcal{F}$ is the total force. If this total is constant, finite source-delay corrections can still create higher acoustic multipole sources.

For a nonseparating small body in a smooth nearly inviscid eddy of speed $U$ and length $\ell$, the fluctuating resultant is an added mass or pressure-gradient force $F\sim {\rho }_0{a}^3{U}^2/\ell$. A uniform pressure integrates to zero over a closed surface. With time $\ell /U$, its acoustic dipole power divided by the eddy's acoustic quadrupole power is $(a/\ell {)}^6/M^2$, where $M=U/c_0$ is the Mach number. Equal powers therefore give the displayed threshold. An area-scaled separated drag force with the same eddy time instead gives $a/\ell \sim \sqrt{M}$; a body-scale time gives a different criterion. The threshold depends on the specified loading and time models.

Thickness noise is the sound generated by displacement of fluid by a moving impermeable body, represented by the surface acoustic monopole term. For a rigid body, ${\int }_S{v}_{n}dS=d\mathcal{V}/dt=0$, so its leading acoustic compact-source approximation has no net-volume monopole. Its local source need not vanish: a moving finite-volume body can radiate through higher spatial moments. A thin-body or negligible-volume approximation is needed when all such radiation is to be omitted.

In the incompressible near field of a spherical bubble, $u_r=a^2\dot{a}/s^2$. Its volume-flux acoustic monopole is the displayed term at retarded time. The integrated surface pressure force vanishes by spherical symmetry, while the Lighthill stress tensor approximation ${\rho }_0{u}_i{u}_j$ has moment $S_{ij}=4\pi {\rho }_0{a}^3{\dot{a}}^2{\delta }_{ij}/3$. Its leading compact acoustic quadrupole contribution is ${\rho }_0(a^3{\dot{a}}^2{)}^{\prime \prime }/(3c_0^{4}r)$. For small fractional radius oscillation $\delta$ and compactness parameter $\alpha =\omega a_0/c_0$, its size relative to the leading monopole is $O(\delta {\alpha }^2)$.

For a small rigid sphere translating with velocity $\mathbf{V}(t)$, the integrated thickness source ${\rho }_0\int \mathbf{V}\cdot {\mathbf{n}}_{s}dS$ vanishes. The first spatial moment is instead ${\rho }_0\int y_j\mathbf{V}\cdot {\mathbf{n}}_{s}dS={\rho }_0\mathcal{V}{V}_j$, where $\mathcal{V}$ is the sphere volume. Expanding the retarded time across the compact source gives the displayed first nonzero thickness field. It is an acoustic dipole, at the same order as acoustic loading noise for a translating sphere. The added mass of a sphere supplies half this volume-mass coefficient through the loading term; their sum is $3{\rho }_0\mathcal{V}\mathbf{n}\cdot \ddot{\mathbf{V}}/(8\pi c_0^{3}r)$.

This tensor contains convective momentum flux, departure from the reference linear pressure-density relation, and viscous stress. Its double divergence is an acoustic quadrupole distribution in Lighthill acoustic analogy.

Write a piecewise smooth Lighthill stress tensor as $T_{ij}=T_{ij}^{-}+D_{ij}H(S)$, where $D_{ij}=T_{ij}^{+}-T_{ij}^{-}$ and the regular level surface $S=0$ is a shock wave. The distributional derivative of the Heaviside step function givesHere repeated indices are summed, $S_i={\partial }_{i}S$ and $S_{ij}={\partial }_i{\partial }_{j}S$. Keep the coefficients as smooth extensions before multiplying distributions: prematurely replacing the coefficient of ${\delta }^{\prime }(S)$ by its surface trace can lose a $\delta (S)$ contribution. The extra surface delta distributions and their derivatives are supported entirely on the discontinuity.