- 1 r (1 - cos 0) = (1) 



CO = A / -2- 



. . F = co It = V y R 



WAVES 419 



and on substituting this value and simplifying we have 



: 



i.e., in deep water the velocity is independent of the height of wave and of 

 the depth of water, and depends only on the length of wave. This is true, 

 within small limits, so long as the depth of water is greater than one-half 

 the length of the wave. 



In shallow water the velocity of propagation becomes equal to .y ~ - 



where a and b are the horizontal and vertical axes of the elliptical orbit, 

 and vary with the depth of water. In waves which are very long 

 compared with the depth, the velocity approximates to V g h, where h is 

 the depth, so that as the water becomes shallower and shallower the 

 velocity approximates more nearly to that of the wave of translation. 

 The wave of oscillation in shallow water is indeed intermediate between 

 the true wave of oscillation and the wave of translation. 



There is no definite relationship between the length and height of 

 waves. Both depend largely on the velocity of the wind which has raised 

 them, and on the distance travelled under its influence. Where this 

 distance is great the velocity is practically identical with the mean 

 velocity of the wind. 1 For short "fetches," such as are usual in 

 reservoirs, Stevenson gives the formula. 



k = 1-5 V~D + 2-5 - * V~D 



where k is the height of wave in feet, and D is the " fetch " in miles. 

 As deep sea waves approach a shallowing beach, the orbital motion of 

 their lower particles is checked, they partake to an increasing extent of 

 the nature of waves of translation, and finally usually break when the 

 depth of water is little more than the height of the wave. 



ART. 118s. RIPPLES. 



So far it has been assumed that the only forces involved in wave 

 formation and propagation are those due to the masses of the particles of 



* Dr. Vaughan Cornish, ''Cantor Lecture," 1912. 



