446 Mr. stokes, ON THE THEORY OF OSCILLATORY WAVES. 



The general value of (p given by (13), which is derived from (2) and (3), must now be restricted to 

 satisfy (17). It is evident that no new terras in (p involving sin mw or cos mw need be introduced, 

 since such terms may be included in the first approximate value, and the only other term which can 

 enter is one of the form B {e""^'"'''' + f-'"''"'") sin 2 m.r. Substituting this term in (17), and 

 simplifying by means of (14), we find 



= c (£"■* _ £-»"»)■■' ■ 



Moreover since the term in (p containing sin ma; must disappear from (17), the equation (l4) will 

 give c to a second approximation. 



If we denote the coefficient of cos mx in the first approximate value of y, the ordinate of the 

 surface, by o, we shall have 



. go^ ca 



mc{i'^'' + f-"*) €"•* - 6"°^ ' 



and substituting this value of A in that of (p, we have 



<p= -ac ^,„, _ ^_„, sin m.v + 3ma'c ^^mi. _ ^-mi.y sin 2 m j; ... (18). 



1 he ordinate of the surface is given to a second approximation by (g). It will be found that 



„ (e™* + e-"*) (e""" + e-''"'' + 4) 

 y = a cos mx-ma- 2 {e""- - e-'"")' cos2mx (19), 



k 



7- The equation to the surface is of the form 



y = a cos mw - K a^ cos 2m3! (20), 



where K is necessarily positive, and a may be supposed to be positive, since the case in which it is 

 negative may be reduced to that in which it is positive by altering the origin of a; by the quantity 



— or — , \ being the length of the waves. On referring to (20) we see that the waves are sym- 

 metrical with respect to vertical planes drawn through their ridges, and also with respect to vertical 

 planes drawn through their lowest lines. The greatest depression of the fluid occurs when a.' = 



\ 3\ 



or = ± \, &c., and is equal to o - a'K: the greatest elevation occurs when ,?• = ± - or = ± — , &c., 



and is equal to a +d^K. Thus the greatest elevation exceeds the greatest depression by 2a'K. 

 When the surface cuts the plane of mean level, cos mw - aKcos 2mx = 0. Putting in the small 



term in this equation the approximate value m.v = - , we have cos mx = - aK = cos I — \- aK\ 



whence .t = ± ( — + ) , = ± f — + 1 , &c. We see then that the breadth of each hollow, 



V4 2 7r / V 4 2 7r / 



x n IT \ 



measured at the height of the plane of mean level, is - + , while the breadth of each elevated 



r u ii J • ^ «-^^ 

 portion 01 the fluid is . 



^ 2 TT 



It is easy to prove from the expression for K, which is given in (ly), that for a given value 

 of X or of m, K increases as h decreases. Hence the difference in form of the elevated and 

 depressed portions of the fluid is more conspicuous in the case in which the fluid is moderately 

 shallow than in the case in which its depth is very great compared with the length of the waves. 



