198, 199] WAVES IN SHALLOW WATER. 535 



If h is infinite this reduces to 145), while if the depth is very small 

 with respect to the wave-length, it reduces to a? = gh. Accordingly 

 long waves in shallow water travel with a velocity independent of 

 their length, being the velocity acquired by a body falling through 

 a distance equal to one -half the depth of the water. Consequently 

 the resultant of such waves of different wave-lengths is propagated 

 without change, contrary to what is the case in deep water. 

 Changing to fixed axes, we have for the running wave 



- *(A0 - -*(A i n & ( x _ a f), 



and by comparison with 147), 150), we find that the particles 

 describe ellipses with semi -axes equal to 



161) G (e*( A +y)-|- e-*(A+?)), C ( e *(A+y)-_ g- *(*+?)). 



If we consider the resultant of two equal wave -trains running 

 in opposite directions, we have 



sfc(# at) -f cos ~k (x -f at)] 

 = 2 (7 (*<*+*>+ e-^+y^cosJcxcosJcat, 



sin k (x - at} -I- sin k(x + at)] 



The equation of the profile is now of the form, y is equal to a 

 function of x multiplied by a function of tf, so that the profile is 

 always of the same shape, with a varying vertical scale. Such waves 

 are called standing waves, and we see them in a chop sea. The 

 difference between them and the stationary wave in a running stream, 

 with which we began, is very marked, as here every point on the 

 surface oscillates up and down, while there the water -profile was 

 invariable both as to time and place. 



199. Equilibrium Theory of the Tides. We shall now 

 briefly consider some aspects of the phenomena of the tides, the 

 general theory of which is far too complicated to be dealt with here. 

 The earliest theory historically is that proposed by Newton, which 

 supposes that the water covering the earth assumes, under the attraction 

 of a disturbing body, the form that it would have if at rest under 

 the action of the forces in question. This so-called equilibrium 

 theory, which neglects the inertia of the water, belongs logically to 

 the subject of hydrostatics, but will be now treated. If U denote 

 the potential of gravity, including the centrifugal force, as in 149, 

 we have, as there, for the undisturbed surface of the ocean, 



