540 XI. HYDRODYNAMICS. 



If there is no disturbing force, X = 0, and we have the equation 



for the propagation of free waves, which we might have used in 

 order to obtain the results of 198, for instance it is satisfied by 

 equations 150) if we put a 2 = gh. This is the same equation as we 

 had in 46, equation 109), for the motion of a stretched string, 

 and the standing waves of 162), 198, putting y = 0, are the normal 

 vibrations of equation 115), 46. The general solution of equation 182) 

 is obtained in the next section, for the present it is sufficient to 

 consider the wave already obtained which advances unchanged in 

 form with the velocity a. We have then, in the case of an endless 

 canal encircling the earth, the curvature of which we may neglect, 

 the case of a free wave, running around and around, without change, 



so that at any point, the motion is periodic in the time ; where I 



is the length of the endless canal. We thus have a system with 

 free periods, and when we consider the action upon it of periodic 

 disturbing forces, we may expect the phenomena of resonance, as 

 described in Chapter Y. 



Let us now suppose the canal coincides with a parallel of 

 latitude, and that x is measured to the westward from a certain 

 meridian. We then have for the horizontal component of the 

 disturbing force 



where V is given by 170), and H, the hour -angle of the moon at 

 the point x, is 



x dV cV 1 



183) H=at > so that ~ = ^^ - > 

 rcosi|) ex dHrcoaip 



03 being the angular velocity of the moon with respect to the earth. 

 We accordingly find X to be composed of two terms each of the form 



where for the semi-diurnal part 



184) A = -jjj- cos2 ^ cos tyj m 2&, fr = 

 Introducing this into the equation 181), 



185) r4 = & 2 o-| ^.sin(w Jcx). 



Ot VX 



we may find a solution 



