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

Tangential phase matching givesA transmitted propagating wave exists when its normal wavenumber is real, equivalentlyFor incidence angles between zero and $\pi /2$, this is automatic if $c_P^{\prime }\le c_P$ and imposes the corresponding critical-angle bound if $c_P^{\prime }>c_P$.

Suppressing the common tangential phase, suitable displacement fields areandFor an inviscid elastic liquid, $\mu =0$, $c_P^2=\lambda /\rho$, and $\mathbf{\sigma }=\lambda (\nabla \cdot \mathbf{u})I$. Continuity of normal displacement and normal traction at $z=0$ givesTogether with the phase-matching relation from part (i), these determine $R$ and $T$.

Eliminating $T$ from the two boundary conditions givesThe Snell law for elastic and acoustic waves impliesTherefore the displacement reflection from an interface between two inviscid elastic liquids is

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.