Wave Refraction Angle Calculator

Refraction is the turning of wave crests caused by depth-dependent wave speed. Because shallow-water celerity is sqrt(g d), the part of a crest over shallower ground travels slower and pivots toward it, so waves progressively align with the depth contours. Over straight parallel contours the geometry is identical to optics and obeys Snell law: sin(alpha2) / sin(alpha1) = C2 / C1. A 10 s wave has a deep-water celerity of 15.6 m/s, but at 5 m depth it slows to 7.0 m/s, so an approach of 30 degrees in deep water reduces to about 13 degrees at that contour. Height changes through two coefficients applied to the deep-water height: the refraction coefficient Kr = sqrt(cos alpha1 / cos alpha2), which reflects the spreading or convergence of wave rays, and the shoaling coefficient Ks = sqrt(Cg0 / Cg), which reflects the slowing of energy transport. Refraction is what makes headlands erode and bays accumulate sand. Rays converge on a submarine ridge or promontory, pushing Kr above 1.0 and concentrating energy, while over a submarine canyon or inside a bay they diverge and Kr falls below 1.0, producing the sheltered conditions that let sediment settle. Breaking follows once height reaches roughly 0.78 of local depth, the saturated surf-zone limit used here. The main limitations are structural: Snell law in this form assumes straight, parallel, gently sloping contours and no wave-current interaction, and it cannot represent diffraction around breakwaters, reflection, or ray crossing over irregular bathymetry. Where rays cross, ray-tracing predicts infinite height and a spectral model must replace the analytical solution.

The optical analogy came first. Willebrord Snellius established the refraction law for light around 1621, and Pierre de Fermat later derived it from a least-time principle; because both light and long water waves obey the same eikonal approximation for slowly varying media, the identical relation applies to wave rays over changing depth. The water-wave side rested on George Biddell Airy, whose 1841 linear theory supplied the depth-dependent celerity that makes refraction possible, and on later shallow-water analysis showing that celerity depends only on depth once the bed is close enough to the surface. Refraction became a working coastal engineering procedure in the 1940s. Sverdrup and Munk at Scripps applied refraction diagrams to convert deep-water swell forecasts into surf conditions for wartime amphibious landings, and Robert Arthur, Walter Munk and John Isaacs published a graphical ray-construction method in 1952 that let engineers draw wave rays across a bathymetric chart by hand. Refraction diagrams built this way explained observed erosion at headlands and the wave shadows behind offshore banks, and they were standard practice in the US Army Corps of Engineers Shore Protection Manual from its 1973 edition. Hand construction gave way to computed ray tracing in the 1970s and then to phase-averaged spectral models, notably SWAN, released by Delft University of Technology in the mid-1990s, which handles refraction together with generation, dissipation and nonlinear transfer.