Wave Celerity Shallow Deep Calculator

Celerity is the speed of an individual wave crest, and for small-amplitude surface gravity waves it follows from the dispersion relation omega squared = g k tanh(k d), where k = 2 pi / L. The hyperbolic tangent splits the problem into regimes. When depth exceeds about half a wavelength, tanh(k d) is within one percent of unity and celerity reduces to C = gT / (2 pi), roughly 1.56 T metres per second, with wavelength 1.56 T squared; a 10 s swell runs at 15.6 m/s with a 156 m wavelength. When depth falls below about one twentieth of a wavelength, tanh(k d) approaches k d and celerity becomes C = sqrt(g d), depending only on depth. Conventional cut-offs are d/L above 0.5 for deep water and below 0.05 for shallow, with an intermediate band between. Energy travels at the group velocity Cg = n C, where n runs from 0.5 in deep water to 1.0 in shallow water, so crests in a deep-water group appear at the rear and vanish at the front. Because energy flux must be conserved as a wave shoals, the shoaling coefficient Ks = sqrt(Cg0 / Cg) first dips slightly below 1.0 in intermediate depths before rising, and height grows until breaking near H of about 0.78 times depth. The theory is linear: it assumes small steepness, no current and a gently sloping bed. Steep or near-breaking waves need Stokes or cnoidal theory, and an opposing current shortens and steepens waves, with blocking possible once the current reaches about one quarter of the deep-water celerity.

The mathematics here comes from the linear wave theory George Biddell Airy published in his article Tides and Waves, written for the Encyclopaedia Metropolitana and issued in 1841. Airy linearised the free-surface boundary conditions for an inviscid, irrotational fluid of constant depth and obtained the dispersion relation with its hyperbolic tangent, together with the deep and shallow limits still taught today. George Gabriel Stokes extended the analysis to finite amplitude in 1847, producing the higher-order series that carries his name and explaining the sharpened crests and flattened troughs of real waves. Joseph Boussinesq in 1872 and Diederik Korteweg with Gustav de Vries in 1895 developed the shallow-water equations for solitary and cnoidal waves, formalising the wave John Scott Russell chased along the Union Canal in 1834. The theory became an engineering tool during the Second World War, when Harald Sverdrup and Walter Munk at Scripps built the first quantitative surf and swell forecasting method for Allied amphibious landings, published openly in 1947; Charles Bretschneider extended it in the 1950s into the SMB curves. The US Army Corps of Engineers codified the results in the Shore Protection Manual, first issued in 1973, and later in the Coastal Engineering Manual. Because the dispersion relation is implicit in wavelength, explicit approximations were developed for computation, notably those of Carl Eckart in 1952, John Hunt in 1979, and John Fenton with W. D. McKee in 1990, the last accurate to well under one percent across all depths.