The long-wave paradox in the theory of gravity waves

Abstract

The theory of long waves in shallow water under gravity employs two different approaches, which have given rise to a well-known paradox remarked by Stokes, but hitherto not fully resolved. On the one hand it was shown by Airy (1) that, if the pressure at any point in the fluid is equal to the hydrostatic head due to the column of water above it, then no wave form can be propagated without change hi shallow water of uniform depth; on the other hand, it was shown by Rayleigh's theory (8) of the solitary wave that this conclusion may be incorrect. In the present paper an attempt is made to elucidate the paradox. Waves of small amplitude η0 and large horizontal wave-length λ (compared with the depth h of the water) are studied, and it is shown that Airy's conclusion is valid if η0λ2/h3 ia large, whereas the solitary wave has η0λ2/h3 of order unity. Equations of motion are derived corresponding to large, moderate and small values of η0λ2/h3; these can be summarized in a single equation for the profile η(x, t):due to Boussinesq (3).It is also shown that the linearized theory of surface waves is valid only if η0/λ and η0λ2/h3 are both small. Some remarks are made on the generation of the solitary wave, and on the breaking of waves on a shelving beach.