Acoustic reflection and transmission at an interface

For two inviscid media, continuity of normal displacement and normal stress determines the reflected and transmitted longitudinal amplitudes.

The transmitted-to-incident amplitude ratio at an interface, with physical variable specified. At normal incidence, stress amplitudes give $T_{\sigma }=2Z_b/(Z_a+Z_b)=1+R_{\sigma }$, while particle velocity gives $T_v=2Z_a/(Z_a+Z_b)=1-R_{\sigma }$. The transmitted energy fraction is $4Z_a{Z}_b/(Z_a+Z_b{)}^2$; amplitude greater than one therefore does not by itself violate conservation of energy.

Fix a positive normal from medium 1 toward medium 2, and define all velocity amplitudes in that coordinate. A right-going linear sound wave satisfies $p^{\prime }=Z{u}^{\prime }$ and a left-going wave satisfies $p^{\prime }=-Z{u}^{\prime }$, with acoustic impedance $Z=\rho c_s$. Continuity gives $u_i+u_r=u_t$ and $Z_1(u_i-u_r)=Z_2{u}_t$. Eliminating $u_t$ gives the displayed reflected-to-incident velocity ratio. The pressure reflection coefficient is its negative. A surface mass, surface restoring force or oblique incidence changes the boundary calculation.

Equal normal velocities and the membrane tension-mass balance determine complex pressure amplitudes. With identical fluids on both sides the reflected and transmitted squared amplitudes sum to one, expressing conserved mean energy flux.

Equating the absolute reflected and transmitted displacement amplitudes, then using acoustic Snell's law, gives an algebraic condition on incidence angle and the two moduli-over-speed impedances.