By Gordon David Crapper
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Extra resources for Introduction to Water Waves
Or This will change slowly in time, the increase in time being 58 ox1(ak1jat)ot/2n. [ Ch. e. 132). 137) ) ak1 a�<;z / (2. 138) w as a function of say, since and are slowly varying functions of Xi_, and does not depend on t but in general does) . Of course also depend on but it is helpful to keep this variation separate. Then we can write from and In Xz and are held constant, whereas in Xz, are held constant. ' g is The left-hand side here is a total derivative, not now moving with a particle, but with the group velocity.
Trochoid. a a -- -- Fig. 69) with a �, A 2n. (b) The linear sinusoidal wave, the first term of (2. 69) with the same a, A. 5 COMPLEX VARIABLE METHODS Although Stokes originally used the expansion method of the last section he came to realise that there is a simpler approach using complex variable theory. 7 1) u = -aya l/1 v = aa xl/1 , where as usual u and v are velocity components. 72) 44 Solutions for Small Waves [Ch. e. streamlines. In a steady flow the free surface does not change in time and therefore has to be a stream line with flow along it.
K 'lt sin xox x1 - ! k cosxo x x2 = rr/2 + Nrr (2. 1 20) the amplitude vanishes completely. These node lines are perpendicular to the crests. In general it is easy to show that the short crested pattern = a x1 + l2 x2) cos(k1x1 + kzx2 -wt) ¢ = aw-m coshm(y+h) mh cos(l1x1 + l2Xz) sin (k1x1 + kzx2 -wt) 17 cos(4 · sinh . (2. e. k and I are perpendicular. This is left as an exercise for the reader. Notice particularly the change in the dispersion relation, replacing on the right. Short crested gravity waves in deep water -+ 00) travel at speed times the speed of plane waves with the same wave length, that is the short crested waves are faster.