450 



Mr. stokes, on THE THEORY OF OSCILLATORY WAVES. 



remaining are the last three, and it will easily be found that their sum is equal to m^a^c" sin mj;, so 

 that (25) becomes 



g (j>, - c^ 0" + m^d'c'' sin mx = (26). 



The value of (p will evidently be of the form ^ «-""." sin w.r. Substituting this value in (2()). 

 we have 



(w^c- - mg) A + m^c?& = 0. 



Dividing by mA, and putting for A and c" their approximate values -ac, — respectively in 



m 

 the small term, we have 



m& - g -^ nva^g, 



"(S)'(-4»-) = (a'(-'-^)' 



The equation to the surface may be found without difficulty. It is 



y = a coimx - ^ma? cos9.m.v + ^rrv'a' coi 3 m, v*, (27) : 



we have also A; = 0, ^ = - oc (1 - | rn'o') e"*"* sin ma;. 



The following figure represents a vertical section of the waves propagated along the surface 



of deep water. The figure is drawn for the case in which a = — . The term of the third order 



80 



in (27) is retained, but it is almost insensible. The straight line represents a section of the plane 

 of mean level. 



13. If we consider the manner in which the terms introduced by each successive approximation 

 enter into equations (7) and (8), we shall see that, whatever be the order of approximation, the 

 series expressing the ordinate of the surface will contain only cosines of mo! and its multiples, 

 while the expression for <j) will contain only sines. The manner in which y enters into the 

 coefficient of cos rmx in the expression for (p is determined in the case of a finite depth by 

 equations (2) and (3). Moreover, the principal part of the coefficient of cos rmx or sin rmv will 

 be of the order a' at least. We may therefore assume 



<f> = 2,' a"-^, (£""(*-*) + 6-™'*-*») sin rmw, 

 y = acosma; + %,' a''B,cosrma!, 

 and determine the arbitrary coefl^cients by means of equations (7) and (8), having previously 

 expanded these equations according to ascending powers of y. The value of c- will be determined 

 by equating to zero the coeflficient of sin mx in (7). 



Since changing the sign of a comes to the same thing as altering the origin of x by ^ \, it is 

 plain that the expressions for A^, B^ and c^ will contain only even powers of a. Thus the values 

 of each of these quantities will be of the form C^ + C^ar + C^a* + ... 



It appears also that, whatever be the order of approximation, the waves will be symmetrical with 

 re.spect to vertical planes passing through their ridges, as also with respect to vertical planes 

 passing through their lowest lines. 



• It is remarkable that this equation coincides with that of the 

 |>rolate cycloid, if the latter equation be expanded according to 

 ascending powers of the distance of the tracing point from the 

 centre of the rolling circle, and the terms of the fourth order be 

 omitted. The prolate cycloid is the form assigned by Mr. Rus. 



sell to waves of the kind here considered. Reports of the BritUh 

 Association, Vol. vi. p. 448. When the depth of the fluid is not 

 great compared with the length of a wave, the form of the surface 

 does not agree with the prolate cycloid even to a second approx- 

 imation . 



