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Acoustic sources may be organized by the number of spatial derivatives in their effective wave equation forcing. A direct scalar source is an acoustic monopole, a divergence of a force density an acoustic dipole, and a double divergence of a stress tensor an acoustic quadrupole.

For a low-Mach number source with stress $O({\rho }_0{U}^2)$ and advective time $\ell /U$, the three-dimensional compact acoustic quadrupole has density amplitude $O({\rho }_0{M}^4\ell /r)$, where $M=U/c_0$. The power depends on these source-time and stress hypotheses, not only on the quadrupole label.

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.

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 ${\partial }_t(\rho u_i)$:Set $q=p-{\hat{p}}_0$ and $W_{ij}=\rho u_i{u}_j-{\sigma }_{ij}$. Since $p+\chi =q+P$, the constant $P$ has zero Laplacian. The prescribed reference mass density and reference speed of sound are independent of time, soCombining the two identities givesOnly conservation of mass and conservation of momentum have been used; no equation of state or energy equation was needed. Spatial variation of ${\hat{c}}_0$ 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: ${\hat{\rho }}_0={\rho }_0$ and ${\hat{c}}_0^2=c_0^2$ outside the flow, with $P$ chosen so ${\hat{p}}_0=p_0$ 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 $p^{\prime }=c_0^2{\rho }^{\prime }$ 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.

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.

Apply the moving-interface conservation jump identity first to mass, then to each momentum component. Write $j_i=\rho u_i$ and ${\Pi }_{ij}=\rho u_i{u}_j+p{\delta }_{ij}-{\sigma }_{ij}$, with all quantities understood piecewise on the two sides. Define the jumps of flux relative to the moving shock wave byThe global distributional conservation equations areDifferentiate the first in time and subtract the divergence of the second. With ${\rho }^{\prime }=\rho -{\rho }_0$ and the piecewise Lighthill stress tensor $T_{ij}=\rho u_i{u}_j+(p^{\prime }-c_0^2{\rho }^{\prime }){\delta }_{ij}-{\sigma }_{ij}$, this givesDerivatives act on the complete distributions, including their moving support. The acoustic quadrupole term represents momentum-stress fluctuations throughout the volume. The time derivative of the surface mass-flux defect is an acoustic monopole, representing injection or removal of mass/volume. The divergence of the surface momentum-flux defect is an acoustic dipole, representing a force sheet. This is the distributional acoustic analogy across a moving interface.

For an actual freely propagating fluid shock wave with no singular mass or momentum supply, the Rankine-Hugoniot conditions give $Q=0$ and $L_i=0$. Such a shock does not acquire independent monopole and force-sheet sources merely because it is discontinuous. Its effects remain in the distributional derivatives of $T_{ij}$, including singular derivatives of its jump. Nonzero surface sources are appropriate for an interface with exchange/forcing or for a formulation that omits one side of the fluid.

A shock does not, by itself, justify retaining the $m^4$ scaling. That estimate required a low-Mach number stress $O({\rho }_0{U}^2)$ varying on the slow time $\ell /U$. Fast shock motion, short time scales, or thermodynamic deviations can invalidate that estimate. If these same compact, slow-source assumptions remain valid for the integrated Lighthill stress tensor, its quadrupole estimate still follows, even distributionally. There is no universal replacement power deducible from the mere presence of a shock; nor should vanished physical flux defects be treated as additional independent radiation sources.

In two dimensions, the outgoing harmonic Helmholtz equation kernel has magnitude proportional to $(k_{0}r{)}^{-1/2}$. Combining this spreading with the compact acoustic quadrupole source and $k_0\ell =O(M)$ changes the density-amplitude Mach number power to $7/2$ when geometric range is separated.