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For point injection of constant power, the outward energy flux through a sphere tending to the source must tend to $\dot{E}$. In the adiabatic superbubble similarity solution, this requiresImposing finite regular central pressure and density with $h=O(\eta )$ would instead give zero flux and fail to supply the prescribed power. A resolved source region or an inner contact-boundary model can replace the point-source condition.

If a unit incident plane wave has scattered field ${\psi }_s=e^{ikR}{f}_{\infty }(\hat{\mathbf{r}},{\hat{\mathbf{r}}}_0)/R+O(R^{-2})$, the differential wave scattering cross-section isIt is the far-field limit of $R^2$ times the scattered-to-incident energy flux ratio. The common physical amplitude normalization cancels when the incident and scattered waves are in the same background medium. The formula assumes that the $1/(4\pi )$ Green-function factor, if present, is already included in the far-field pattern.

For a random complex wave amplitude $E$, its mean squared amplitude is $I=\mathbb{E}[|E{|}^2]$. Expanding $E=m+(E-m)$, with $m=\mathbb{E}[E]$, givesThe mixed terms vanish. Thus the squared coherent field is only one part of the mean intensity; the other is the diffuse contribution. A fixed acoustic or optical normalization converts this squared-amplitude quantity to physical energy flux. This ensemble mean differs from the angularly averaged radiative-transfer mean intensity.

For constant isothermal sound speed, time-independent shearing-sheet tidal potential ${\Phi }_t$, and inviscid flow, define $E=\Sigma u^2/2+c_s^2\Sigma \ln (\Sigma /{\Sigma }_{\mathrm{ref}})+\Sigma {\Phi }_t$. The barotropic energy density and kinetic energy balances combine to give ${\partial }_{t}E+\nabla \cdot [(E+P)\mathbf{u}]=0$. Coriolis acceleration does no work. Integral conservation requires vanishing net boundary energy flux.

Write $D_t={\partial }_t+\mathbf{u}\cdot \nabla$ for the material derivative. For any specific quantity $f$, conservation of mass givesDotting the momentum equation with $\Sigma \mathbf{u}$ produces the kinetic energy balanceThe Coriolis acceleration does no work because $\mathbf{u}\cdot ({\mathbf{e}}_z\times \mathbf{u})=0$. Since the shearing-sheet tidal potential is time-independent, its advected potential-energy density obeysFor the isothermal equation of state, define the barotropic energy density $U=c_s^2\Sigma \ln (\Sigma /{\Sigma }_{\mathrm{ref}})$, with fixed positive reference surface density ${\Sigma }_{\mathrm{ref}}$. The continuity equation impliesAdding these balances combines the two pressure terms into $-\nabla \cdot (P\mathbf{u})$. This establishes isothermal shearing-sheet energy conservation. Thus the conserved energy density and its flux areChoosing the density unit so that ${\Sigma }_{\mathrm{ref}}=1$ reproduces the printed logarithm. Changing the reference adds a multiple of the conserved mass to $E$ and its advective energy flux. The barotropic energy density is the mathematical energy of this fixed-isothermal sound speed closure; it is not the microscopic thermal energy of a thermally isolated gas. Maintaining an isothermal process can require heat exchange.

For $\mathbf{r}=R\hat{\mathbf{r}}$ with the observation distance much larger than the scatterer, expand $|\mathbf{r}-{\mathbf{r}}^{\prime }|=R-\hat{\mathbf{r}}\cdot {\mathbf{r}}^{\prime }+O(R^{-1})$. The far-field pattern in the Born approximation for scalar wave scattering is consequentlyThis convention includes the factor $1/(4\pi )$ in $f_{\infty }$; moving that factor into the outgoing-wave definition would change the pattern's normalization.

The absolute energy flux depends on what physical amplitude $\psi$ represents. For acoustic pressure in a homogeneous fluid of mass density ${\rho }_0$ and wave speed $c_0$, the momentum equation gives the particle-velocity amplitude $\mathbf{v}=\nabla \psi /(i{\rho }_0\omega )$. The time average of harmonic power and acoustic intensity then giveThe radial derivative of the outgoing factor is $(ik-1/R)e^{ikR}/R$. Its imaginary current is therefore $k|f_{\infty }{|}^2/R^2$ to leading order. The incident unit-amplitude plane wave has imaginary current $k{\hat{\mathbf{r}}}_0$. Hence, with $k=\omega /c_0$,The scattered energy flux points along $\hat{\mathbf{r}}$ at leading order, and the incident energy flux along ${\hat{\mathbf{r}}}_0$. If $\psi$ is a velocity-potential amplitude instead, both leading expressions use the common prefactor ${\rho }_0\omega k/2$ in place of $k/(2{\rho }_0\omega )$. A normalized scalar current uses $k$ as the common prefactor. The paper does not specify this amplitude normalization; the ratio in part (c) is independent of that choice.

Use the same physical normalization for both fields. Part (b) has $E_s=E_i|f_{\infty }^{(1)}{|}^2/R^2+O(R^{-3})$, so the common energy flux prefactor cancels. The differential cross-section for scalar wave scattering isThe far-field pattern has dimensions of length in this normalization, so the differential cross-section for scalar wave scattering has dimensions of area per unit solid angle. Only the scattered field is used in this ratio; interference between the incident and scattered fields is a different contribution to the total field's energy flux.