At the plmige point the combination of height, bottom slope, depth, 

 and period reaches a limit for a stable wave form and the crest plunges. 



Considering the progress of a single wave train of miiform height 

 and length from deep water to the shore over a gradually sloping 

 bottom, it seems reasonable to assume that the same period would be 

 observed at all points along the path. This statement is by no means 

 axiomatic, for waves often coalesce or break up into shorter waves; 

 however, laboratory experiments on smaU waves moving over limited 

 reaches show that the individual wave crests maintain their identity. 

 The point raised is important in the application of wave equations 

 to field conditions. 



From the theoretical standpoint Airy concluded that the conditions 

 of constant pressure and continuity could not be satisfied simulta- 

 neously on a sloping bottom, and from experiment it is known that an 

 abrupt change in depth may cause waves to break up into shorter 

 waves. In groups including a finite number of waves, as in the wave 

 pattern around a ship, it is observed that waves at the front of the 

 train disappear while new waves develop at the rear. A frequently 

 observed similar phenomenon, where a wind wave can be followed 

 for a considerable distance and then seems to disappear, probably 

 results from the superposition of wave trains of different periods. 

 The fundamental question is whether the component waves of each 

 wave train maintain their identity. If they do, it follows that their 

 period is the same at every point along the path even though the 

 apparent period of the combined wave trains may vary from point to 

 point. 



The question of permanence of oscillatory waves has been the 

 subject of much speculation and it is of interest to review some of the 

 conclusions drawn. Rayleigh (1) stated that: 



* * * I think that the reader who follows the results of the calculations 

 here put forward is likely to be convinced that permanent waves of moderate 

 height do exist. 



It may be mentioned that most of the authorities * * * express belief 

 in the existence of permanent waves, even though the water be not deep, provided 

 of course that the bottom be flat. 



Stokes came to a similar conclusion as follows: 



In fact it will presently appear that it is only an indefinite series of waves which 

 possesses the property of being propagated with uniform velocity and without 

 change of form; at least this is the case when the waves are such as can be propa- 

 gated along the surface of a fluid which was previously at rest. 



It appears from the above, that of all waves for which the motion is in two 

 dimensions, which are propagated in a fluid of uniform depth, and which are 

 such as could be propagated in a fluid previously at rest, there is only one partic- 

 ular kind, namely, that just considered, which possesses the property of being 

 propagated with a constant velocity and without change of form; so that a solitary 

 wave cannot be propagated in this manner. 



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