= Solution
At each point the <Gaussian process> has a standard normal <marginal distribution>. Its <characteristic function> gives the coherent screen field
$$
m=\langle e^{i\beta W(z)}\rangle=e^{-\beta^2/2}.
$$
It is constant in $z$, so its <Fourier transform> is $m\delta(\nu)$. Applying the <Fresnel propagator> to this zero-transverse-<wavenumber> component gives \b[the ensemble-averaged reduced spectrum]
$$
\boxed{\langle\widehat E(x,\nu)\rangle
=e^{-i\nu^2x/(2k)}e^{-\beta^2/2}\delta(\nu)
=e^{-\beta^2/2}\delta(\nu).}
$$
Thus the reduced mean is independent of $x$. The physical mean $\langle\psi(x,z)\rangle=e^{ikx}e^{-\beta^2/2}$ still has its carrier phase. The <coherent attenuation by a Gaussian phase screen> takes place at the screen, with no further coherent attenuation in homogeneous space.
The full <spectral acoustic flux> depends on the second moment, not the square of this mean. For a <stationary process>, the generalized cross-spectral correlation is
$$
\langle\widehat E(x,\nu)\overline{\widehat E(x,\nu')}\rangle
=\delta(\nu-\nu')S_E(\nu).
$$
Indeed its propagation factors cancel on $\nu=\nu'$. This explains why the ensemble-averaged <spectral acoustic flux> in (b) is also independent of $x$, even for nonzero transverse <wavenumbers> whose individual complex amplitudes change phase. The coherent atom has weight $|m|^2=e^{-\beta^2}$; the full <stationary phase-screen power spectrum> also contains fluctuations. Under the decay assumptions in (b), their integrated weight is $1-e^{-\beta^2}$. One must use these weights in a <spectral measure of a stationary random field>, rather than square a <Dirac delta distribution>. A phase-only screen preserves the leading-order total <acoustic energy flux>, although it reduces its coherent part.
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