Wave scattering at a planar interface

Wave scattering at a planar interface separates an incident plane wave into reflected and transmitted plane waves. Translation invariance parallel to the interface conserves the tangential component of the wavevector.

Relative to a distinguished propagation coordinate $z$, the vertical wavenumber is the corresponding wavevector component $q$. For a plane wave of wavenumber $k$ making angle $\theta$ with the vertical, $q=k\cos \theta$.

Snell's law follows from conservation of tangential wavenumber across a planar interface. If angles are measured from the interface normal, it reads $n_1\sin {\theta }_1=n_2\sin {\theta }_2$.

For a circular index-$n$ tank, an eye at $(\sqrt{2}R,0)$, and a swim diameter $x=0$, trace outward single-pass rays through $P=R(\cos \varphi ,\sin \varphi )$, with $|\varphi |\le \pi /4$. Backward refraction gives the dimensionless intercept $y=\sin \varphi -\cos \varphi \tan (\varphi -i)$, where $\sin i=\sqrt{2}\sin \varphi /[n\sqrt{3-2\sqrt{2}\cos \varphi }]$. The visible fraction is the largest positive intercept. For $n=1.33$ it is approximately $0.75744$, attained before the grazing outgoing ray. The limiting pair of stationary optical paths merges at a caustic.

For a scalar field satisfying second-order wave equations with the same leading coefficient on either side of an interface, continuity of the field and its normal derivative supplies two interface conditions. Different constitutive laws can instead impose weighted normal-derivative continuity.

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.