Solution
= Solution
Tangential phase matching gives
$$
k\sin\theta=k'\sin\theta',
\qquad
\boxed{\frac{\sin\theta}{c_P}
=\frac{\sin\theta'}{c'_P}}.
$$
A transmitted propagating wave exists when its normal wavenumber is real, equivalently
$$
\boxed{\left|\frac{c'_P}{c_P}\sin\theta\right|\leq1}.
$$
For incidence angles between zero and $\pi/2$, this is automatic if $c'_P\leq c_P$ and imposes the corresponding critical-angle bound if $c'_P>c_P$.