 Normal-incidence acoustic velocity reflection coefficient

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.