Past exam of the mathematics course of the University of Cambridge / 2015 / iii / Paper 76 / 2 / d

Let the medium's autocorrelation function of a random field and power spectrum use the explicit conventionSince $n^2-1=2\mu W+{\mu }^2{W}^2$, only the linear term in $\mu$ is needed to obtain the cross-section through second order:Squaring and taking the expectation givesThis is the finite-volume formula in the Born approximation for scalar wave scattering; the zero mean of $W$ does not make its squared scattered amplitude vanish. For a jointly centered Gaussian random field, the third-order contribution is zero, so the next perturbative contribution is fourth order when that expansion is valid.

Introduce the overlap volume $H_V(\mathbf{s})=|V\cap (V-\mathbf{s})|$. Changing variables rewrites the double integral asWhen the linear dimensions of a regular scattering volume are large compared with the correlation length, and $C$ is integrable, $H_V(\mathbf{s})\simeq |V|$ over the significant covariance range. This gives the finite-volume random scattering spectrum resultThere are two distinct approximations here: the weak-scattering expansion in $\mu$ and the negligible-boundary-overlap approximation in the volume-to-correlation-scale ratio. If $S$ instead denotes the power spectrum of the index fluctuation $n-1=\mu W$, then $S={\mu }^2{S}_W$ and the external factor ${\mu }^2$ is absorbed into $S$.

Statistical homogeneity alone allows an anisotropic spectrum, so its argument is generally the vector ${\mathbf{k}}_s$. If statistical isotropy is additionally assumed, $S_W$ depends only onThe scalar notation $S(k_s)$ is valid in that isotropic case. For the alternative convention $S_{\mathrm{norm}}=(2\pi {)}^{-3}{S}_W$, the bulk formula is $2\pi {\mu }^2{k}^4|V|S_{\mathrm{norm}}({\mathbf{k}}_s)$. Changing the sign of the scattering vector does not change the spectrum of a real stationary scalar field; changing the Fourier transform normalization does change the displayed $2\pi$ factors. Both conventions have therefore been stated explicitly.