= Solution
No reflection requires
$$
\lambda'\sin2\theta=\lambda\sin2\theta'.
$$
Using $\lambda=\rho c_P^2$, $\lambda'=\rho'(c'_P)^2$, and
$\sin\theta'=(c'_P/c_P)\sin\theta$, this reduces to the normal <acoustic impedance> matching condition
$$
\rho'c'_P\cos\theta=\rho c_P\cos\theta'.
$$
After squaring and eliminating $\theta'$, the required incidence angle satisfies
$$
\boxed{
\sin^2\theta=
\frac{\rho'^2(c'_P)^2-\rho^2c_P^2}
{(c'_P)^2(\rho'^2-\rho^2)}
}.
$$
Such a no-reflection angle exists only when the right-hand side lies in $[0,1]$ and the transmitted wave is propagating.
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