= Solution
Use the prescribed harmonic convention $e^{i\omega t}$. For a propagating <acoustic plane wave>, let $k=(\omega/c_0)\cos\theta$ and $\ell=(\omega/c_0)\sin\theta>0$. The incident and reflected <pressure> amplitudes in the upper half-space have vertical factors $e^{i\ell y}$ and $e^{-i\ell y}$ respectively:
$$
p'=e^{i\omega t-ikx}\left(Ae^{i\ell y}+Re^{-i\ell y}\right).
$$
The <linear homentropic acoustic equations> imply $i\omega\rho_0v_y=-\partial_y p'$. At the surface, the <normal velocity> is therefore $\ell(R-A)/(\rho_0\omega)=-V$, while the <pressure> amplitude is $P=A+R$. The <surface acoustic impedance> condition $P=ZV$ gives
$$
\boxed{\frac{A+R}{A-R}=\frac{Z\sin\theta}{\rho_0c_0},\qquad
\frac RA=\frac{Z-Z_f}{Z+Z_f},\quad Z_f=\frac{\rho_0c_0}{\sin\theta}.}
$$
Here $Z_f$ is the <normal acoustic impedance>; the angle in this question is measured from the horizontal, not the normal. For a <passive acoustic impedance>, the mean power absorbed per unit area is $\tfrac12\operatorname{Re}(PV^*)=\tfrac12\operatorname{Re}Z\,|V|^2\geq0$. The four limiting cases have distinct meanings:
* If $Z\to0$, $R=-A$. This is a <pressure-release boundary>: the <pressure> perturbation vanishes, while the <normal velocity> is generally nonzero. The reflected <pressure> has equal amplitude and a phase reversal.
* If $Z\to\infty$, $R=A$. The boundary is <acoustically rigid>, with zero <normal velocity> and doubled total surface <pressure>. There is no <pressure> phase reversal.
* If $R/A\to0$, $Z\to Z_f$. This is a matched boundary, taking up the incoming wave without reflection. Its <pressure> and <normal velocity> are those of the incident wave.
* Formally, $R/A\to\infty$ means $Z\to-Z_f$. A nonzero outgoing field can then exist with vanishing incoming amplitude. For a passive boundary at a real propagating incidence angle, a negative-real-part <surface acoustic impedance> cannot describe ordinary absorption: such a scattering pole must be interpreted through an active source or an continued by <analytic continuation> free-mode resonance. The sheet calculation below identifies the relevant free modes.
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