258 Lord Rayleigh on Waves. 



the pipe, the fluid which approaches an enlargement must lose 

 velocity, and the change of momentum involves an augmented 

 pressure. On this account, therefore, there is an increased 

 pressure at a place of enlargement and a diminished pressure 

 at a contraction. On the other hand the effect of gravity is in 

 the opposite direction, tending to produce a loss of pressure at 

 the upper surface where that surface is high, and a gain of 

 pressure where the surface is low. This effect of gravity is 

 independent of the velocity; but the changes of pressure due to 

 acceleration and retardation depend on the velocity of flow, 

 and we can therefore readily understand that, with a certain 

 definite velocity of flow, compensation may take place, at least 

 approximately. When this happens, the condition of a free 

 surface is satisfied, the constraint may be removed, and we are 

 left with a stationary wave-form. 



In the theory of long waves it is assumed that the length is 

 so great in proportion to the depth of the water, that the velo- 

 city in a vertical direction can be neglected, and that the hori- 

 zontal velocity is uniform across each section of the canal. 

 This, it should be observed, is perfectly distinct from any sup- 

 position as to the height of the wave. If I be the undisturbed 

 depth, and h the elevation of the water at any point of the 

 wave, u , u the velocities corresponding to l,l + h respectively, 

 we have, by the condition of continuity, 



lu 



"= T+k' 



so that 



By the principles of hydrodynamics, the increase of pressure 

 due to retardation will be 



2 V ° J 2 (Z + A) 2 * 



On the other hand, the loss of pressure due to height will be 

 gph ; and therefore the total gain of pressure over the undis- 

 turbed parts is 



9p] h - 



r 2 1 + 



* l d + 



K) 



If now the ratio h : I be very small, the coefficient of h becomes 



