Reflection and transmission coefficients at a scalar-wave interface ($R,T$)

For incident and transmitted normal wavenumbers $q_0$ and $q_1$, continuity of a scalar field and its normal derivative gives

The reflection coefficient is the complex ratio between reflected and incident wave amplitudes. Its modulus determines amplitude attenuation and its argument determines the phase shift on reflection.

The reflected-to-incident compressional pressure or normal-stress amplitude ratio at a planar interface. For normal incidence between lossless homogeneous media, the displayed expression uses acoustic impedances $Z_j={\rho }_j{V}_{pj}$. The reflected energy fraction is $r_p^2$, while a same-coordinate particle-velocity reflection coefficient has the opposite sign.

The transmission coefficient is the complex ratio between transmitted and incident wave amplitudes. Flux transmission may also contain an impedance or normal-wavenumber factor and therefore need not equal $|T{|}^2$.

The transmitted-to-incident compressional pressure or normal-stress amplitude ratio. At normal incidence, continuity of normal stress gives $t_p=1+r_p$, with pressure-amplitude reflection coefficient $r_p$. The transmitted energy fraction is $(Z_1/Z_2)t_p^2$, not $t_p^2$ alone.