= Solution
For the incident <acoustic plane wave>, set $q_0=k\sin\theta$ and $\beta_0=k\cos\theta>0$. The flat reflected field and total field are
$$
\psi_s^{[0]}=-e^{iq_0x+i\beta_0z},\qquad \psi^{[0]}=e^{iq_0x}(e^{-i\beta_0z}-e^{i\beta_0z}).
$$
Therefore $\partial_z\psi^{[0]}(x,0)=-2i\beta_0e^{iq_0x}$ and the <first-order rough-surface scattered field> is
$$
\psi_s^{[1]}(x,z)=\frac{2i\beta_0}{2\pi}\int\widehat h(q-q_0)e^{iqx+i\beta(q)z}dq.
$$
It is linear in the height. Since $\langle h(x)\rangle=0$, its mean vanishes, with the expectation interpreted through finite windows or stationary spectral distributions when needed. Hence
$$
\boxed{\langle\psi_s(x,z)\rangle_{\text{through first order}}=-e^{ik(x\sin\theta+z\cos\theta)}.}
$$
\b[The coherent first-order reflection is the flat-surface reflection.] If the field symbol is instead used only for the rough correction, its first-order mean is zero. <Stationarity> ensures the coherent reflection retains the incident horizontal wavenumber, but zero mean height already explains the vanishing linear correction. The nonzero <root mean square> height does not enter this mean at first order; it does enter the fluctuating reflected field and its intensity.
Back to article page
