Past exam of the mathematics course of the University of Cambridge / 2022 / ii / Paper 2 / 40C / c / iv / Solution

No reflection requiresUsing $\lambda =\rho c_P^2$, ${\lambda }^{\prime }={\rho }^{\prime }(c_P^{\prime }{)}^2$, and
$\sin {\theta }^{\prime }=(c_P^{\prime }/c_P)\sin \theta$, this reduces to the normal acoustic impedance matching conditionAfter squaring and eliminating ${\theta }^{\prime }$, the required incidence angle satisfiesSuch a no-reflection angle exists only when the right-hand side lies in $[0,1]$ and the transmitted wave is propagating.