On the formation of water waves by wind (second paper)

In a previous paper I investigated the problem of the formation of waves on deep water by wind, and found that the available data were consistent with the hypothesis that the growth of the waves is due principally to a systematic difference between the pressures of the air on the front and rear slopes. Lamb had already discussed the maintenance of waves against viscosity by an approximate method, but without obtaining numerical results. Being under the incorrect impression that Lamb’s approximation would not hold for the short waves I was chiefly considering, I proceeded on more elaborate lines. It now appears, however, that Lamb’s method is not only applicable to the problem of waves on deep water, but is readily extended to cover the case when the water is comparatively shallow, and to allow for surface tension. The fundamental approximations are first, the usual one that squares of the displacements from the steady state can be neglected, and second, that viscosity modifies the motion of the water to only a small extent. The motion of the water can then, to a first approximation, be considered as irrotational. With the previous notation, let ζ be the elevation of the free surface x, y, z the position co-ordinates, t the time, U the undisturbed velocity of the water, h the depth, and φ the velocity potential. Also let σ, p, q , and ϑ denote respectively ∂/∂ t , ∂/∂ x , ∂/∂ y , and ∂/∂ z , and write p 2 + q 2 = - r 2 .