68 Generation, Growth and Propagation of Waves 



of wave formation is far ahead of observations and needs to be checked by 

 experiments. 



2. Propagation of a Wave Disturbance Through a Region Previously Undisturbed 



Before going more thoroughly into the theory of the generation and growth 

 of water waves, it is advisable to know something about the propagation 

 of the waves which, after being generated at a certain point, advance from 

 this source in a given direction. The theoretically very difficult memoirs by 

 Poisson (1816, p. 71) and Cauchy (1827) give the nature of such a pro- 

 pagation. A disturbance is produced on the surface of an infinitely large 

 water mass of great depth. The problem now is to find out how the disturbance 

 changes the form of the surface and how does the disturbance propagate. 

 We can imagine that this disturbance is composed of an infinite number 

 of sine and cosine waves of all possible wave lengths, according to Fourier's 

 Theorem, and we can discuss the further development of this package of waves. 



The formulation of the problem can be accomplished in two ways: (1) we 

 start with an initial elevation of the free surface without initial velocity; (2) we 

 start with an undisturbed surface (and therefore horizontal) and an initial 

 distribution of surface pressure impulse. 



Every kind of disturbance can be brought back to a combination of these 

 two basic kinds. The results from the theory are essentially identical for 

 both kinds of initial disturbances, so that it will be sufficient to deal only 

 with one of them. In doing so, it will be well to distinguish between the cases 

 in which, the propagation of the disturbance takes place only in one direction 

 (canal waves) or occurs in all directions (circular waves). The results are 

 not essentially different, so far as the form and the velocity of the propagation 

 are concerned. Therefore, we will only concern ourselves with canal waves. 

 The theoretical developments of these cases can be found in Lamb's Hydro- 

 dynamics (1932, p. 384). The disturbance at a point x = in the canal is 

 confined to the immediate neighborhood of the origin. If b is the width of 

 the canal and F the area between the disturbed and the undisturbed water 

 surface, then Fb will be the volume of water originally lifted up above the 

 surface and causing the disturbance by falling back to the level of the undis- 

 turbed surface. It will then cause a change in the level at a far distant point 

 which has the form 



rj = -y—[C(w)cosw+S(w)smw] , (IV. 1) 



X f 71 



with 



V 2 



v. 



W ■ 4.v 



C(w) and S(w) are the integrals of Fresnel, which are of great importance 

 in the theory of the diffraction of light. Tables of these integrals can be found 

 in Jahnke and Emde (1933). When the quantity vv is larger, i.e. after 



