Past exam of the mathematics course of the University of Cambridge / 2023 / ii / Paper 2 / 40C (Waves)

Let $k_S=\omega /c_S$ and $k_P=\omega /c_P$. Snell law for elastic waves fixes the reflected waves' common tangential wavenumber:Choose the reflected SV-wave and converted reflected P-wave asThe polarizations are respectively perpendicular and parallel to their wavevectors. Phase matching makes all three exponential factors equal at $z=0$. The rigid boundary condition for an elastic wave ${\mathbf{u}}_I+{\mathbf{u}}_{RS}+{\mathbf{u}}_{RP}=0$ there givesSolving this two-by-two linear system yieldsThese fields give the Reflection of an SV-wave from a rigid plane.

The reflected P-wave is propagating only while its normal wavenumber is real, equivalently while $|\sin \varphi |\le 1$. Sinceit is evanescent whenPutThen its spatial factor iswhich decays as $z\to -\infty$. The reflected SV amplitude becomesThe numerator and denominator have equal complex modulus, soThus evanescent mode conversion changes the reflected SV phase but not its amplitude, as described by Unit-modulus SV reflection with evanescent P conversion.

Average the instantaneous elastic-wave energy flux over one temporal period at fixed position. For complex amplitudes,Write the evanescent P displacement amplitude asFor an isotropic linear elastic material with Lamé parameters $\lambda ,\mu$, its stress and velocity amplitudes areApart from a real common factor,which is purely imaginary. Its real part vanishes, and henceThe converted P-wave is a reactive near-boundary field and transports no time-averaged acoustic energy in the normal direction.