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An acoustic analogy rearranges fluid conservation equations into a chosen linear acoustic propagation operator plus effective forcing. The rearrangement can be exact before source approximations are made. A term classified as forcing can also contain propagation effects omitted from the reference operator; this produces propagation-source ambiguity in an acoustic analogy. The Lighthill acoustic analogy is a familiar homogeneous reference-medium example.

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.

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.

Use the conservative continuity equation and momentum balance, with ${\sigma }_{ij}$ the viscous stress:Differentiate the first equation in time and take the divergence of the second. Eliminating momentum givesSubtract $c_0^2\Delta \rho$, and define the Lighthill stress tensor byThe constant reference subtraction has no double divergence. The exact Lighthill acoustic analogy isThis is an exact rearrangement; linearization is not used to eliminate the nonlinear source tensor.

Convolving with the retarded acoustic Green function and transferring the source derivatives by integration by parts givesThe acoustic compact-source approximation requires source size $\ell$ to satisfy $\Omega \ell /c_0\ll 1$ for its characteristic frequency $\Omega$; the acoustic far field additionally requires $r\gg \ell$ and $\Omega r/c_0\gg 1$. Internal retardation is then negligible at leading order, and the radiating parts of the two observer derivatives act on the common retarded time. With $n_i=x_i/r$ and $S_{ij}=\int T_{ij}{d}^{3}y$, the far-field acoustic force and stress moments giveThe two spatial derivatives supply two factors of $1/c_0$, in addition to the $1/c_0^2$ in the Green function.

For a subsonic fluctuating source of velocity scale $V$, size $\ell$ and turnover time $\ell /V$, suppose its stress scale is $T_{ij}=O({\rho }_0{V}^2)$. Then $S_{ij}=O({\rho }_0{V}^2{\ell }^3)$ and ${\ddot{S}}_{ij}=O({\rho }_0{V}^4\ell )$. The compact acoustic quadrupole Mach-number scaling is thereforeThis fourth-power statement concerns the acoustic density amplitude under the specified eddy-time and stress scaling. With an independently imposed frequency, the general estimate instead retains $V^2{\Omega }^2{\ell }^3/c_0^{4}r$.

Use mass conservation and conservative momentum conservation, with viscous stress ${\sigma }_{ij}$ defined so that its divergence is the viscous force:Differentiate the first equation in time and eliminate the momentum time derivative using the second. This givesSet ${\rho }^{\prime }=\rho -{\rho }_0$ and $p^{\prime }=p-p_0$, where the reference values and $c_0$ are constant. Subtract $c_0^2{\nabla }^2{\rho }^{\prime }$ to obtain the exact Lighthill acoustic analogyThe Lighthill stress tensor includes nonlinear momentum transport, departure from the reference linear pressure-density relation, and viscous stress. For an inviscid fluid the last term is absent. The derivation is an exact rearrangement of the conservation equations; treating the tensor as a known localized source is a subsequent approximation.

Use time dependence $e^{i\omega t}$ and write the outgoing two-dimensional Helmholtz equation kernel as $G_2\sim C{e}^{-i{k}_{0}r}/\sqrt{k_{0}r}$, where $C$ is independent of the Mach number. The harmonic Lighthill acoustic analogy givesFor a compact acoustic quadrupole, the source integral is $O({\rho }_0{U}^2{\ell }^2)$. In the radiation region each derivative of the outgoing exponential supplies a factor of $k_0$, soKeeping the same advective source-frequency scaling as in (i), $\omega =O(U/\ell )$ and hence $k_0\ell =O(m)$. Therefore the two-dimensional compact quadrupole scaling isThe changed exponent comes from cylindrical spreading $(k_{0}r{)}^{-1/2}$. Compactness $k_0\ell \ll 1$ and the far-field condition $k_{0}r\gg 1$ are separate assumptions. At fixed imposed $\omega$ and fixed $\ell$, with only $U$ varied, the stress instead gives an $O(m^2)$ dependence; monochromaticity alone does not supply the additional source-frequency assumption.

Put $r=|\mathbf{x}|$, $\mathbf{n}=\mathbf{x}/r$, and take a compactly supported Lighthill stress tensor. The isotropic term in that tensor is $(p-c_0^2\rho ){\delta }_{ij}$: the Kronecker delta is implicit in the printed shorthand. Convolving the Lighthill acoustic analogy with the retarded acoustic Green function and integrating twice by parts in the source coordinates givesThe boundary terms vanish because of compact support. For source size $\ell$, the acoustic compact-source approximation requires $\omega \ell /c_0\ll 1$. At $r\gg \ell$, we may therefore replace the denominator by $r$ and the retarded argument by $t-r/c_0$ in the leading source integral. Define $S_{ij}(t)=\int T_{ij}(\mathbf{y},t)d^{3}y$. In the radiation region $\omega r/c_0\gg 1$, the leading two spatial derivatives act on the retarded time rather than on $1/r$ or $\mathbf{n}$:Thus the leading acoustic quadrupole density isFor the compact acoustic quadrupole Mach-number scaling, let $U$ be the source fluctuation speed, $m=U/c_0\ll 1$, and use the advective source time $\ell /U$. With $T_{ij}=O({\rho }_0{U}^2)$, one has $S_{ij}=O({\rho }_0{U}^2{\ell }^3)$ and ${\ddot{S}}_{ij}=O({\rho }_0{U}^4\ell )$. ConsequentlyThe fourth power refers to the radiation coefficient with geometric spreading separated. It assumes that the source frequency scales as $U/\ell$; an independently imposed frequency or an independently strong thermal source would change that Mach number estimate.

Let $R=|\mathbf{x}|$, $\mathbf{n}=\mathbf{x}/R$ and $t_R=t-R/c_0$. Convolving the source terms in the Lighthill acoustic analogy with the retarded acoustic Green function, and moving their spatial derivatives outside the integral, givesThe integrals have compact spatial support. For an acoustically compact source, its size $l$ satisfies $\omega l/c_0\ll 1$, so the source-dependent retardation is negligible at leading order. Also take $R\gg l$. Define $S_{ij}=\int T_{ij}{d}^{3}y$ and $F_i=\int f_i{d}^{3}y$. In the radiation terms the spatial derivatives act on $t_R$, with ${\partial }_{x_i}{t}_R=-n_i/c_0$. Derivatives of $1/R$ produce smaller near-field terms. Thus the far-field acoustic force and stress moments giveUsing $n_i=x_i/R$ gives the equivalent powers of $R$ requested. The force term is an acoustic dipole; the Lighthill stress tensor produces an acoustic quadrupole.

For a small body in low-speed flow, put $M=u/c_0\ll 1$. A typical unsteady force has size $F=O({\rho }_0{u}^2{l}^2)$ and time scale $l/u$, whereas $S=O({\rho }_0{u}^2{l}^3)$. ThereforeThese are the acoustic dipole Mach-number scaling and compact acoustic quadrupole Mach-number scaling. Since $p^{\prime }=c_0^2{\rho }^{\prime }$ and outgoing acoustic intensity is of order $p^{\prime 2}/({\rho }_0{c}_0)$, their sound powers scale asThus an unsteady force on the body can produce the dominant low-Mach number sound. If the integrated force or its time derivative vanishes, that dipole contribution is absent. Force on the body and force on the fluid have opposite signs.

Use complex amplitudes with time dependence $e^{i\omega t}$, so the outgoing two-dimensional Helmholtz equation kernel has phase $e^{-i{k}_{0}r}$. Dividing the harmonic form of Lighthill acoustic analogy by $c_0^2$ and integrating the two source derivatives by parts gives an amplitude proportional toOnly the magnitude is needed here; the specified kernel omits its constant phase and normalization. In the acoustic compact-source approximation, $\int {\tilde{T}}_{ij}{d}^{2}y=O({\rho }_0{U}^2{\ell }^2)$. With $\omega =O(U/\ell )$, hence $k_0\ell =O(m)$, the two-dimensional compact quadrupole scaling isThe half-power difference from three dimensions comes from cylindrical spreading, including its $k_0^{-1/2}$ factor. As in part (i), the Mach number power is stated with the geometrical range factor separated; $k_{0}r\gg 1$ is still required.

Let $r=|\mathbf{x}|$, $n_i=x_i/r$, and let the source have size $\ell$. The retarded acoustic Green function gives the outgoing solution of Lighthill acoustic analogy asThis follows by integrating the source derivatives by parts in the retarded convolution. Assume the source is localized and the boundary terms vanish. In the acoustic compact-source approximation, $k_0\ell \ll 1$, the retardation across the source can be neglected, while $r\gg \ell$ permits replacement of the denominator by $r$. The integral becomes $S_{ij}(t-r/c_0)/r$.

In the radiation region $k_{0}r\gg 1$, derivatives of the retarded argument dominate derivatives of the spreading factor. Since ${\partial }_{x_i}(t-r/c_0)=-n_i/c_0$, the leading acoustic quadrupole field isThe two negative retardation derivatives give a positive sign. This is a far-field approximation, with smaller near-field terms omitted.

For low-Mach number aerodynamic fluctuations of speed $U$ and advective time $\ell /U$, take $T_{ij}=O({\rho }_0{U}^2)$, $S_{ij}=O({\rho }_0{U}^2{\ell }^3)$, and two time derivatives of order $(U/\ell {)}^2$. ThereforeThe quoted fourth power is the compact acoustic quadrupole Mach-number scaling, with geometric spreading shown explicitly. It assumes the source strength and time scale just stated; the source must be acoustically compact, and the observation point must remain in the radiation region.