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Full text of "Scientific Papers - Vi"

308                            DEEP WATER WAVES,  PBOGRESSIVE  OB                           [393
iri which I, V, c, c', are small compared with a, a'.    Thus (6) gives
, .     , (   d*a .         d2a'         \    dz{3 .   0 (1  a cos so   sm x)(  r^ sin % + -, cos x    -^ sin J# v                                / \    at2              at2         J     at*
,   .     N /da.              do.'  .     \     r/,                      /       \
- (a cos a 4- a sm a?) ( -^ cos as + -?r sm x ]  {(1 - a cos a?  a sm a?)
\ flC                      66              /
x (a sin so  a cos x) + 2/3 sin 2fc  2/3' cos 2a + 87 sin 3  87' cos 3
x         -r               j
dt              dt
This equation is to hold good to the second order for all values of x, and therefore for each Fourier component separately. The terms in sin* and cos x give
*                            *               A                             /fu
0 ...................... (9)
The term in sin 2as gives and, similarly, that in cos 2$ gives
In like manner
and so on.    These are the results of the surface condition Dp/Dt = 0.    From the other surface condition (p = 0) we find in the same way
'
,    d{3    a dot.'    a da.
From equations (9) to (16) we see that a, a' satisfy the same equations (9) as do a, a', and also that c, c' satisfy the same equations (12) as do 7, 7'; but that b, V are not quite so simply related to /3, /3'.. (6)