Lord Rayleigh on Waves. 259 



and we conclude that the condition of a free surface is satisfied 

 provided u 2 Q —gl. This determines the rate of flow in order 

 that a stationary wave may be possible, and gives of course at 

 the same time the velocity of a wave in still water. 



If we suppose the condition u\—gl satisfied, the change of 

 pressure is, to a second approximation, 



j . f + 21 ,\ 3rf 



H) 



which shows that the pressure is defective at all parts of the 

 wave where h differs from zero. Unless, therefore, h 2 can be 

 neglected, it is impossible to satisfy the condition of a free sur- 

 face for a stationary long wave — which is the same as saying 

 that it is impossible for a long wave of finite height to be pro- 

 pagated in still water without change of type. If, however, h 

 be everywhere positive, a better result can be obtained with 

 an increased value of u ; and if h be everywhere negative, 

 with a diminished value. We infer that a positive wave moves 

 with a somewhat higher, and a negative wave with a somewhat 

 lower velocity than that due to half the undisturbed depth. 



Although a constant gravity is not adequate to compensate 

 the changes of pressure due to acceleration and retardation in 

 a long wave of finite height, it is evident that complete com- 

 pensation is attainable if gravity be a function of height ; and 

 it is worth while to inquire what the law of force must be in 

 order that long waves of unlimited height may travel with type 

 unchanged. If/ be the force at height h, the condition of con- 

 stant pressure is 



2 o 



whence 



{i-^j^JV; 



f== _ u ° d p ±<* l * 



2 dh(l + h) 2 °(l + h) 1 



which shows that the force must vary inversely as the cube of 

 the distance from the bottom of the canal. Under this law the 

 waves may be of any height, and they will be propagated un- 

 changed with the velocity \/fl, where / is the force at the un- 

 disturbed level. 



The same line of thought may be applied to the case of a 

 long wave in a canal whose section is uniform but otherwise 

 arbitrary. Let A be the area of the section below the undis- 



T2 



