419.
ON PERIODIC IRROTATIONAL WAVES AT THE SURFACE OF DEEP WATER.
[Philosophical Magazine, Vol. XXXIIL pp. 381—389, 1917.]
THE treatment of this question by Stokes, using series proceeding by ascending powers of the height of the waves, is well known. In a paper with the above title* it has been criticised rather severely by Burnside, who concludes that " these successive approximations can not be used for purposes of numerical calculation...." Further, Burnside considers that a numerical discrepancy which he encountered may be regarded as suggesting the non-existence of permanent irrotational waves. It so happens that on this point I myself expressed scepticism in an early paperf, but afterwards I accepted the existence of such waves on the later arguments of Stokes, MuCowanJ, and of Korteweg and De Vries§. In 19111| I showed that the method of the early paper could be extended so as to obtain all the later results of Stokes.
The discrepancy that weighed with Burnside lies in the fact that the value of {3 (see equation (1) below) found best to satisfy the conditions in the case of a=^ differs by about 50 per cent, from that given by Stokes' formula, viz. ft = — ^a\ It seems to me that too much was expected. A series proceeding by powers pf £ need not be very convergent. One is reminded of a parallel instance in the lunar theory where the motion of the moon's apse, calculated from the first approximation, is doubled at the next step. Similarly here the next approximation largely increases the numerical value of /9. When a smaller a is chosen (•£$), series developed on Stokes' plan give satisfactory results, even though they may not converge so rapidly as might be wished.
The question of the convergency of these series is distinct from that of the existence of permanent waves. Of course a strict mathematical proof of their existence is a desideratum; but I think that the reader who follows the results of the calculations here put forward is likely to be convinced that
* Proc. Lond. Math. Soc. Vol. xv. p. 26 (1915).
t Phil. Mag. Vol. i. p. 257 (1876); Scientific Papers, Vol. i. p. 261.
J Phil. Mag. Vol. xxxn. pp. 45,, 553 (1891).
§ Phil. Mag. Vol. xxxix. p. 422 (1895).
|| Phil. Mag. Vol. xxi. p. 183 (1911). [This volume, p. 11.]ore battery contact is made. The breaking of this contact introduces a constraint, and the charge on the first conductor is reduced. In all such problems potential corresponds with force and charge corresponds with displacement.