= Solution
For the given Gaussian radial spectrum, the only needed <integral> is
$$
\int_0^\infty \mu^2L^3e^{-\nu^2L^2/4}\nu\,d\nu=2\mu^2L.
$$
If $S$ is interpreted with the self-reciprocal Hankel pair in the supplied hint, the coherent field is
$$
\boxed{m(x,\mathbf r)=e^{-k^2\mu^2Lx}(U_xE_0)(\mathbf r).}
$$
In this convention $B_n(r)=2\mu^2L e^{-r^2/L^2}$ and hence $B_n(0)=2\mu^2L$, directly verifying the attenuation coefficient. If instead the stated spectrum is the unnormalized Cartesian <Fourier transform>, use its inverse normalization and obtain
$$
\boxed{m(x,\mathbf r)=e^{-k^2\mu^2Lx/(2\pi)}(U_xE_0)(\mathbf r).}
$$
Its transverse <covariance> coefficient is then $B_n(r)=\mu^2L\,e^{-r^2/L^2}/\pi$. The apparent difference is precisely the inconsistent Fourier/Hankel normalization in the printed hint, not a difference in stochastic propagation. For a unit constant entrance field the Fresnel factor is one; coherent intensity is the square of the displayed mean amplitude.
Back to article page
