On the formation of water waves by wind

Abstract

It is well known that in certain circumstances a type of instability may arise at the surface of separation of two fluids when there is a finite difference between the velocities on the two sides of the surface. Some disturbances of the surface, of simple harmonic type, may increase exponentially in amplitude until the customary simplifying assumption, that the terms of the second degree in the displacements from the undisturbed state can be ignored, breaks down. One would naturally expect that in the case, for instance, of a wind blowing over the surface of water, waves would be first formed when the velocity of the wind is just great enough to make one particular type of wave grow; thus the critical wind velocity and the wave-length of the waves first formed will constitute checks on any theory of wave formation. The problem for frictionless fluids has been solved by Lord Kelvin, subject to the restriction that the disturbances considered are two-dimensional, no horizontal displacement occurring across the relative velocity of the fluids. Since, however, the possible initial deformations of a horizontal surface will not as a rule satisfy this condition, an investigation of the growth or decay of deformations of other types is desirable. 1. Hypothesis of Irrotational Motion . Let the two fluids be incompressible (a legitimate approximation so long as the wave velocity is small compared with that of sound in either fluid) and of great vertical extent. Let the origin be in the undisturbed position of the surface of separation and the axis of z vertically upwards. Let ζ be the elevation of the surface, and suppose the two fluids to have initially velocities U and U' parallel to the axis of x , accents referring to the upper fluid. Let the densities of the fluids be respectively ρ and ρ', and the velocity potentials in them Ф and Ф' . Let the operators ∂/∂ t , ∂/∂ x , ∂/∂ y , ∂/∂ z be denoted by σ, p , q , and ϑ respectively. Putting r 2 for — ( p 2 + q 2 we see that ∇ 2 Φ = 0 (1) is equivalent to (ϑ 2 — r 2 ) Ф = 0, (2) whence Ф = U x + e rz A, (3) where A is a function of x and y , determined by the value of Ф where z is zero.