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In the Born approximation for scalar wave scattering, with $n=1+\mu W$ and zero-mean stationary unit-variance $W$, define $S_W(\mathbf{q})=\int C(\mathbf{s})e^{-i\mathbf{q}\cdot \mathbf{s}}{d}^{3}s$. The leading expectation of the differential cross-section for scalar wave scattering isFor a regular volume much larger than the covariance range, the overlap is approximately $|V|$, giving ${\mu }^2{k}^4|V|S_W({\mathbf{k}}_s)/(4{\pi }^2)$. Statistical isotropy, beyond statistical homogeneity, permits the scalar argument $|{\mathbf{k}}_s|=2k\sin (\theta /2)$. If the spectrum is defined for the index fluctuation rather than $W$, it already contains ${\mu }^2$.

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.