﻿Wave of Permanent Type, 353 



satisfied for a solitary wave, with the exception of the con- 

 dition of constant surface-pressure, are identically satisfied 

 by (1): for w vanishes with z, and when ^=+00, u=- — U 

 and w = 0. We proceed, therefore, to determine the surface- 

 pressure with a view to the determination of the available 

 constants so as to satisfy as nearly as may be this condition 

 for a free surface : as it is, (1) may be regarded as giving a 

 particular forced wave. 



2. The Surface-Pressure near the Mean Level, 

 Expanding (1), and writing for brevity 



«=l + 2/, /3=l-4/, &c, .... (3) 



we get 



— (?.* + cw)/TJ = 1 —JaP sec 2 \m{z + ix) —^PW sec 4 \m(z + ox) ; (4) 

 therefore, further, for x positive 



- (m+ a0)/U= 1-2^6-^-^ + 2{2cc-/3k 2 )k 2 e- 2m (*-^ &c; (5) 

 therefore, integrating with respect to z + ix, 



-m{f + L(j))/II = m(z + ix)+2i*kh- m ( x -^-i(2*-/3k^ (6) 



Now if h be the mean depth, or the depth at an infinite 

 distance from the crest, we must, taking ^=0 at the bottom, 

 have tJt=— UA at the surface. Also, if?/ denote the eleva- 

 tion of the surface at any point above the mean level, we must 

 have z = h + rj at the surface. Substituting these values in 

 the expression for ^ involved in (6) , we obtain as the equation 

 to the surface, 



mrj = 2ak 2 €- mx sin m(h + n) — (2u—/3k 2 )k 2 6- 2mj! sm 2m(h + rj) . . . ; 



or, when rj is small, 



mrj = 2uP6- mx smmh— (2a— /3P — 2a 2 P)Pe- 2nM? sin2raA...; (7) 



which gives the equation to the surface in a convenient form 

 for points not too near the crest. 

 Again, from (5) we get 



q 2 /W = l-4:ccPe- m *cosmz + 4:k 2 6- 2m *{{2cc-pk 2 ) cos 2mz + 4a 2 & 2 } 5 (8) 



therefore, by (7), we have at the surface where rj is small, 



<?7U 2 = 1 — 4:*k 2 e- mx cos mh 



+ U 2 €~ 2mx { {2oc-/3k 2 ) cos 2mh + « 2 & 2 (1 + 2 sin 2 mh) } . . . . (9) 



Now in a liquid of density p moving irrotationally acted 

 on by no force but gravity, the pressure p at any point is 

 given by 



p= constant — tP9 9P z > • • > • (10) 



P/»7. J%. S. 5. Vol. 38. No. 233. Oct. 1894. 2 B 



