On Periodic Irrotational Waves at the Surface of Deep Water

W. Burnside
1917 Proceedings of the London Mathematical Society  
No exact solution of the problem of permanent irrotational waves in deep water, where the water surface merely changes its position in space without changing its form, has hitherto been discovered. Indeed it is not known whether any exact solution exists. In the ordinary treatment of the problem, to a first approximation, it is assumed that all terms containing the square of the velocity function are negligible. On this assumption it is shewn that if the ratio of the height (vertical distance
more » ... vertical distance from trough to crest) to the length of the waves is small enough a sinusoidal wave can be propagated without sensible change. Exactly what " small enough " means the theory as ordinarily presented gives no means of determining. For a wave of given shape, height and length, it should be possible to calculate the extreme variation of pressure over the free surface, and the ratio of this to, say, the hydrostatic difference of pressure between trough and crest gives some idea of the permanence of the motion. In the absence of an exact theory it is difficult to see that more than this can be done. Of the different calculations carrying the algebraic approximation to a high order of small quantities, the most complete is that given by Stokes (Collected Papers, Vol. I, " Supplement to a Paper on the Theory of Oscillatory Waves ")• The approximation is carried to the fifth power of the parameter " b," and to extend it is merely a matter of taking trouble. The complete result would be to express the coordinates x and y in terms of 0 and \fr in the form CO x = -d>+be* sin + 2 b n P n {b)e n * sin n, 2
doi:10.1112/plms/s2-15.1.26 fatcat:srqfi5dtgbcb3drautdsdtwdye