48 Mr. J. M°Cowan on the Solitary Wave. 



By means of (4), (9) is immediately reducible to 



q 2 = U 2 - 2mU(^ + JJz) cot mz + m 2 {f + U;) 2 (cosec 2 m£ - 2 /ma) . (10) 



Let h be the mean depth, or, which is the same thing, the 

 depth at an infinite distance from the wave, then by (4) the 

 value of y{r at the surface will be — JJh. Let rj be the eleva- 

 tion of the surface above the mean level, then at the surface 

 ^ + U: = TJ77, and therefore by (4) the surface will be given 

 by the equation 



sin m{h + rj) , 11x 



r] = a -pr \ ^-, : . . . (11) 



cos in [Ji + 77) + cosh m x 



and for the surface, (10) will reduce to 



q 2 = U 2 { 1 — 2mrj cot in (h + rj) + m 2 ?7 2 (eosec 2 ?n(h + 77) — 2 /ma) . (12) 



So far the equations are exact. Now expanding (12) in 

 powers of 77 and neglecting all beyond rf it becomes 



q 2 = U 2 { 1 — 2mrj cot mh + ))i*rf(o cosec 2 mA — 2 /ma) 



— 4m V cot m/> cosec 2 mh + &c.}; . (13) 



but for the free surface the condition to be satisfied is 



tf = TP-2gi, (14) 



So comparing (13) with this we see that if we take 



mU 2 cot mh=g and Sma = 2 sin 2 mh 



the motion under consideration will satisfy the condition for 

 a free surface to that degree of approximation in which the 

 term containing rf in (13) is considered negligible. 



It is possible, however, to get at once a much closer approxi- 

 mation : for if we take ma = | sin 2 m(Ji + §770), where r} is a 

 quantity of the same order of magnitude as 77, (13) becomes 



q 2 = JJ 2 {1 — 2mrj cot mh + 4:m 3 r) 2 (7} — rj) cot mh cosec 2 mh\, (15) 



and thus the condition (14) will be accurately satisfied 

 (rj 4 &c. neglected) where y = Vo as we ^ as °f C0lirse where 

 97 = 0. Hence, for reasons which will be more fully examined 

 immediately, we shall take for ?7 the maximum elevation of 

 the wave or elevation of the crest above the mean level. 

 Thus, finally, taking 



JJ 2 =gm~ l tan mh (16) 



and 



ma—% sin 2 m(/tH-|77 ), (17) 



where, by (11), 



77 = a tan J m(h + ?7 ), (18) 



