= Solution
The normal <acoustic energy flux> is
$$
I_y=p'u_y=p'\phi_y.
$$
In either half-space, the pressure fields above are proportional to $\cos(kx-\omega t)$, whereas the normal velocities are proportional to $\sin(kx-\omega t)$. They are therefore in temporal quadrature. Averaging over one period gives
$$
\boxed{
\langle I_y\rangle
\mathrel\propto\langle\cos(kx-\omega t)\sin(kx-\omega t)\rangle=0.
}
$$
\b[Thus the field is evanescent and stores <reactive acoustic energy flux> near the membrane without transporting net acoustic energy away from it in the normal direction; the wave's mean energy transport is along the membrane.]
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