JJeep Water Ship- Waves, 737 



which satisfies (65) and expresses a sinusoidal wave-dis- 

 turbance, of wave-length 2wfm } travelling .r-wards with 

 velocity v. 



§ 39. To find the boundary-pressure II, which must act on 

 the water-surface to get the motion represented by {66), when 

 m, w, k are given, we must apply (64) to the boundary. Let 

 r = be the undisturbed surface ; and let d denote its de- 

 pression, at (.r, o, t) below undisturbed level ; that is to say, 



& = %(.v, o, t) = -^-(/>(.r, z, t) g=0 —mksin m(x — vt) (67), 



whence by integration with respect to t, 



k 

 &=~cosm(x—vt) (68). 



To apply (64) to the surface, we must, in az, put ~ = d; and 



in ilcj)/dt we may put z = 0, because d, k, are infinitely small 

 quantities of the first order, and their product is neglected 

 in our problem of infinitesimal displacements. Hence with 

 (66) and (68), and with II taken to denote surface-pressure, 

 (64) becomes 



kmvcosm(x — trt) = -£cosra(#— vt)— II +#0 . (69); 



whence, with the arbitrary constant C taken =0, 



n==ib/^— »U cos m(x-vt) .... (70); 



and, eliminating h by (68), we have finally, 



U = [f,-,nr°~jd (71). 



Thus we see that if v= y/gjm, we have 11=0, and therefore 

 we have a train of free sinusoidal waves having wave-length 

 equal to 2irjm. This is the well-known law of relation 

 between velocity and length of free deep-sea waves. But if 

 e is not equal to s/ f//ni, we have forced waves with a surface- 

 pressure (rj — mv 2 )& which is directed with or against the 

 displacement according as v< or > sjffjm. 



§ 40. Let now our problem be : — given II, a sum of sinu- 

 soidal functions, instead of a single one, as in (70) ; — required 

 d the resulting displacement of the water- surface. We have 

 by (71) and (70), with properly altered notation, 



n=2Bcoswi(j?-irf-f£) (72), 



