A wave theory for sandwaves in shelf-seas

Abstract

Sea-waves are understood more easily if they are considered to move with constant velocity and without changing shape. A convenient mathematical technique is to imagine that these waves can be arrested in space and analysed as for flowing streams with a steady wave-like surface deformation. Hitherto, this technique has been used almost exclusively to study progressive surface waves. For depths as great as 45 m, such wave-like deformations occur on the sea surface as a weakly coupled mirror-image of sandwaves on the seabed. By assuming that these surface deformations are real arrested progressive waves and not merely analogous, we now show that sand-dunes are related empirically by a small-amplitude surface wave dispersion equation. Furthermore, finite-amplitude wave theory (that is, steepest wave theory) is related to the steepest dunes on a stream bed where previously, for want of a better theory, steepest dune wavelengths have been explained in terms of fluid turbulence. In particular, if the dune wavelengths associated here with Gerstner's finite amplitude theory are combined in a manner similar to a beat effect, the predicted wavelengths are in good agreement with the lengths of marine sandwaves.