374 



SURFACE WAVES. 



[CHAP, ix 



wards; thus at a depth of a wave-length the diminution of 

 amplitude is in the ratio e~' 2n or 1/535. The forms of the lines 

 of (oscillatory) motion (^ = const.), for this case, are shewn 

 in the annexed figure. 



In the above investigation the fluid is supposed to extend to infinity in 

 the direction of #, and there is consequently no restriction to the value of /. 

 The formulae also, give, however, the longitudinal oscillations in a canal of finite 

 length, provided k have the proper values. If the fluid be bounded by the 

 vertical planes # = 0, x= I (say), the condition dfyldxQ is satisfied at both ends 

 provided sin^ = 0, or kl = mir, where i = l, 2, 3, .... The wave-lengths of the 

 normal modes are therefore given by the formula \ = 2,l/m. Cf. Art. 175. 



218. The investigation of the preceding Art. relates to the 

 case of ' standing ' waves ; it naturally claims the first place, as a 

 straightforward application of the usual method of treating the 

 free oscillations of a system about a state of equilibrium. 



In the case, however, of a sheet of water, or a canal, of uniform 

 depth, extending to infinity in both directions, we can, by super- 

 position of two systems of standing waves of the same wave-length, 

 obtain a system of progressive waves which advance unchanged 

 with constant velocity. For this, it is necessary that the crests 

 and troughs of one component system should coincide (horizon- 

 tally) with the nodes of the other, that the amplitudes of the two 

 systems should be equal, and that their phases should differ by a 

 quarter-period. 



Thus if we put 77 = % + ^ (1), 



where Vi = a sm &# cos ^> Vz = a cos kx sin crt (2), 



we get 97 = a sin (kx a~t) (3), 



which represents (Art. 167) an infinite train of waves travelling 



