Theory of Short and Long Waves 29 



the decrease of the orbits with increasing depth: the dotted lines represent 

 the successive places of filament of particles which are vertical when they 

 pass through a crest or a trough. The solid drawn lines are the possible wave 

 profiles, which are trochoids, and their extreme form is a cycloid. 



Stokes has already proven that a system of waves of the Gerstner type 

 in an ideal fluid cannot be originated from rest. It can be supposed that by 

 properly adjusted pressures applied to the surface of the waves the surface 

 takes the shape of Gerstner waves. According to Stokes, we require as 

 a generating condition an initial horizontal motion in the direction opposite 

 to that of the propagation of the waves ultimately set up. A wind blowing 

 over a water surface tends to provoke a steady horizontal current in the 

 direction of the waves. Therefore, this is an unfavourable condition for 

 producing Gerstner waves. 



4. Short-Crested Waves 



The previous theoretical investigations assume infinitely long wave crests 

 which advance perpendicularly to the direction of progress of the wave. 

 The surface should look like a sheet of corrugated iron. In reality, however, 

 the ocean surface looks like crepe paper. Jeffreys (1925, 1926) has given 

 the theory for these short-crested waves. We can represent this wave by 



r/ = A cos (at — y.x) cos x'y , (11.23) 



in which the wave advances in the 4-.v-direction with a wave length / = 2nfx, 

 the length of the crest being ).' = 2.t/*\ Figure 16 shows, according to Thorade, 



Fig. 16. Three-dimensional wave motion (// = 7/). (By Thorade.) 



a topographical map of the surface for /.' = 7/.. This presentation is still 

 too regular compared to the actual mixed-up waves as they occur in nature, 

 but there is already a difference with the waves discussed earlier. If x and y 

 are the axes in the horizontal direction and z is positive upwards, we can 

 express the velocity potential by 



tp = — Ae rz s'm(ot — y.x)co$y.'y . (11.24) 



in which r' 2 = x- — v.'-. The amplitude has been selected in such a way that 

 the conditions for the continuity and irrationality are satisfied. We can deduct 



