= Solution
Fix time dependence $e^{-i\omega t}$ and use <pressure> <amplitude> for $\psi$. Write the incident <acoustic plane wave> as $\psi_i=Ae^{i\alpha x-i\beta z}$, where $\alpha=k\cos\theta$, $\beta=k\sin\theta>0$. “Perfectly reflecting” specifies zero absorbed energy but does not uniquely select a boundary condition: a rigid wall has a <Neumann boundary condition>, whereas a <pressure>-release surface has a <Dirichlet boundary condition>. We first give the rigid-wall result, and then the alternative <pressure>-release result.
For the rigid surface the normal-<velocity> condition, without an immaterial normalizing factor, is
$$
(\partial_z-h'(x)\partial_x)\psi(x,h(x))=0.
$$
Expand $\psi=\psi^{[0]}+\psi^{[1]}+\psi^{[2]}+\cdots$ in the height <amplitude>, scaling a fixed differentiable profile. The flat total field is $\psi^{[0]}=Ae^{i\alpha x}(e^{-i\beta z}+e^{i\beta z})$. Taylor expansion gives
$$
\partial_z\psi^{[1]}(x,0)=-h\partial_z^2\psi^{[0]}(x,0)+h'\partial_x\psi^{[0]}(x,0)
=2Ae^{i\alpha x}(\beta^2h+i\alpha h').
$$
Let $\widehat h(q)=\int h(x)e^{-iqx}dx$ and $\beta(q)=\sqrt{k^2-q^2}$ on the outgoing sheet: it is positive real for propagating modes and positive imaginary for evanescent modes. Fourier transforming this boundary derivative gives $2A(k^2-\alpha q)\widehat h(q-\alpha)$. The <outgoing angular spectrum> therefore yields the <small-height Neumann scattering> field
$$
\boxed{\psi_s(x,z)=Ae^{i\alpha x+i\beta z}
+\frac{2A}{2\pi i}\int_{\mathbb R}\frac{k^2-\alpha q}{\beta(q)}\widehat h(q-\alpha)e^{iqx+i\beta(q)z}\,dq+O(h^2).}
$$
For an infinite stationary surface the Fourier quantities may be interpreted as random distributions or through a finite illumination window and its limiting mean. The correction is linear in $h$ and $h'$, so zero mean height and stationarity give
$$
\boxed{\langle\psi_s\rangle=Ae^{i\alpha x+i\beta z}+O(h^2)\quad\text{for a rigid reflector}.}
$$
Only the rough correction has zero mean at first order; the flat reflected wave remains.
The first nontrivial coherent correction depends on correlation, not just the r.m.s. height. Define $C_h(s)=\langle h(x+s)h(x)\rangle$, its <power spectrum of surface height> $S_h(q)=\int C_h(s)e^{-iqs}ds$, and $\sigma^2=(2\pi)^{-1}\int S_h(q)dq$. At second order the flat-field third normal derivative and mixed derivative vanish at zero, leaving
$$
\partial_z\psi^{[2]}(x,0)=-h\partial_z^2\psi^{[1]}(x,0)+h'\partial_x\psi^{[1]}(x,0).
$$
For each first-order outgoing Fourier component, its derivative data are $g_1(q)=2A(k^2-\alpha q)\widehat h(q-\alpha)$, its second normal derivative is $i\beta(q)g_1(q)$, and its tangential derivative is $qg_1(q)/\beta(q)$. Averaging their products with $h,h'$ gives the mean derivative
$$
\langle\partial_z\psi^{[2]}(x,0)\rangle
=-2iAe^{i\alpha x}\int\frac{(k^2-\alpha q)^2}{\beta(q)}S_h(q-\alpha)\,\frac{dq}{2\pi}.
$$
Dividing by the specular outgoing derivative $i\beta$ therefore gives <oblique coherent reflection from a stationary rigid rough surface>:
$$
\boxed{\langle\psi_s\rangle=Ar_Ne^{i\alpha x+i\beta z},\qquad
r_N=1-\frac1{\pi\beta}\int\frac{(k^2-\alpha q)^2}{\beta(q)}S_h(q-\alpha)\,dq+O(h^3).}
$$
This second-order expression requires the relevant spectral moments to exist and the boundary expansion to remain controlled. Small $|kh|$ alone does not control arbitrarily large slopes or high-frequency roughness. The given stationarity and $\sigma$ determine the first-order mean above, but do not determine this spectrum-weighted correction numerically.
For a <pressure>-release surface, instead impose $\psi(x,h)=0$. The flat reflected coefficient is $-1$, and Taylor expansion gives $\psi^{[1]}(x,0)=2i\beta Ahe^{i\alpha x}$. Thus <small-height Dirichlet scattering> gives
$$
\boxed{\psi_s=-Ae^{i\alpha x+i\beta z}
+\frac{2i\beta A}{2\pi}\int\widehat h(q-\alpha)e^{iqx+i\beta(q)z}\,dq+O(h^2).}
$$
The zero-mean first correction again vanishes on averaging. At second order $\psi^{[2]}(x,0)=-h\partial_z\psi^{[1]}(x,0)$, giving
$$
\boxed{\langle\psi_s\rangle=Ar_De^{i\alpha x+i\beta z},\qquad
r_D=-1+\frac\beta\pi\int\beta(q)S_h(q-\alpha)\,dq+O(h^3).}
$$
If the roughness spectrum is concentrated near $q=\alpha$ after the incident tangential shift, the additional long-correlation approximation yields $r_N\simeq1-2\beta^2\sigma^2$ and $r_D\simeq-1+2\beta^2\sigma^2$. A local Gaussian random-height phase model would then give $r_N=e^{-2\beta^2\sigma^2}$ and $r_D=-e^{-2\beta^2\sigma^2}$, but Gaussian statistics and that local approximation were not supplied. They must not be inferred from stationarity and r.m.s. height alone.
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