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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 areTherefore ${\partial }_z{\psi }^{[0]}(x,0)=-2i{\beta }_0{e}^{i{q}_{0}x}$ and the first-order rough-surface scattered field isIt is linear in the height. Since $⟨h(x)⟩=0$, its mean vanishes, with the expectation interpreted through finite windows or stationary spectral distributions when needed. HenceThe 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.

Linearized momentum balance is ${\rho }_0{\partial }_t\tilde{u}=-\nabla \tilde{p}$. For a progressive acoustic plane wave $\tilde{p}=P(\hat{k}\cdot x-c_{0}t)$, integration givesThe integration constant is zero for the pure wave with no added uniform flow. The instantaneous acoustic intensity is $I=\tilde{p}\tilde{u}$, the pressure-work energy flux; time-averaged intensity is $⟨I⟩$. For complex harmonic pressure amplitude $P$, $⟨I⟩=|P{|}^2\hat{k}/(2{\rho }_0{c}_0)$.

Use phase convention $e^{i(k_{y}y-\omega t)}$, with $\omega =k{c}_0$, $k_y=k\sin \theta$, $k_x=k\cos \theta >0$. Let incident, reflected and transmitted pressure amplitudes be $1,R,T_p$. The two normal velocities at the membrane are $(1-R)\cos \theta /({\rho }_0{c}_0)$ and $T_p\cos \theta /({\rho }_0{c}_0)$, and both equal $-i\omega X_0$. Hence $T_p=1-R$ and $X_0=i{T}_p\cos \theta /({\rho }_0{c}_0\omega )$. Only normal velocity is matched for an inviscid fluid; tangential velocities need not agree.

Put $D_m=k^2(m{c}_0^2-T{\sin }^2\theta )$. The dynamic condition is $-D_m{X}_0+T_p-(1+R)=0$, or $-D_m{X}_0-2R=0$. Since $D_m\cos \theta /({\rho }_0{c}_0\omega )=1/\alpha$, elimination givesComplex conjugation gives the amplitudes for the opposite time-phase convention. At zero $D_m$, the formulas have the limiting values $R=0,T_p=1$ even though $\alpha$ itself is then infinite. A strongly inertial membrane has $\alpha \to 0$, giving almost total reflection.

For real $\alpha$, $|R{|}^2+|T_p{|}^2=1$. Thus the incoming net normal energy flux is $\cos \theta (1-|R{|}^2)/(2{\rho }_0{c}_0)$, equal to the transmitted flux $\cos \theta |T_p{|}^2/(2{\rho }_0{c}_0)$. Time-averaged energy flux is conserved: the lossless membrane stores and returns energy but has no mean dissipation.