By Gordon David Crapper
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Additional info for Introduction to Water Waves
The numerical method uses a standard library routine com puter program for systems of ordinary differential equations. 162). Clearly they are remarkably accurate in this simple configuration. 1 Show that
In general cg may be greater than or less than c, but the relative motion is similar. 6w ""* 0 we arrive at the conclusion that the group velocity cg = dw/dk, whereas c = w/k. An interesting point which we shall confirm later is that as the wave amplitude vanishes at the node of the group, energy will not be carried through this point. Wave energy will move with the group, in some sense at speed Cg. wave group A /\. /\ /\. /\. /\. /\ /\. /\ /\. /\ /\. r;/f\--T\-1\:�, ().. \ V\) C\ I Fig. 2 . 1 0 The top sinusoidal profile has the same amplitude as, but longer wavelength than, the middle one.
As we can see by referring back to Fig. but regarding the axis there as the ray. ) where is the angle between k normal to the crests, and Cg parallel to the ray, so that -- 2. 145) ds dt dt We shall usually divide by dN/dt and use N as a new independent variable. It is also useful for some applications to have our equations in polar co ordinates. We define these by r cos(} r sinO in which r :;;;. The wave number has to be resolved into components kr in the radial direction away from the origin and at right angles, as shown in Fig.