30 Theory of Short and Long Waves 



from (11.24) the velocity components u, v, w, and the co-ordinates of a particle 

 referred to the centre of its orbit. 

 This gives: 



I = - Ae rz sm(ot — xx)cosx'y , 

 £' = Ae rz cos {at — xx) sin x'y , 



(11.25) 



ry = Ae rz cos{at — xx)cosx'y . 



If we use again Rayleigh's method by adding a wave velocity —c, we 

 transform the wave motion into a steady motion, and the square of the total 

 velocity of a particle (« — c) 2 + v 2 -\-w 2 = c 2 — 2uc if we neglect u 2 + i?-\-w 2 

 against c 2 . Bernoulli's theorem then is 



i 

 + £»7 + ^ (c 2 — 2wc) = const ; 



which is only possible when g = cxajr. If ajx — c, we obtain 



A 2 



vW 



l, r 



(11.26) 



The velocity of propagation increases when the crests become shorter. 

 We can deduct from (11.25) that the orbits are no longer circular, but elliptical, 

 and the plane of the ellipse is parallel to the direction of propagation (.v-axis), 

 but it forms an angle a with the vertical plane xz. This angle a is given by 

 tana = x'/rtanx'/y. These planes, therefore, are only vertical in the troughs 

 and in the crests; elsewhere they are inclined, and the vertical axis is larger 

 than the horizontal. The ellipses are standing up, which might seem strange. 

 It has not been possible to find out if such waves really exist. Another strange 

 fact is that the energy of such three-dimensional waves per unit area is smaller 

 than for a similar two-dimensional wave. 



When t = 0, we have according to equation (11.23) rj = Acoskxcosk'y. 

 The potential energy of a v/ave crest can be computed by integrating Igqifdxdy 

 from — \X to +JA and from —J )' to +|A' and by dividing afterwards by 

 the area \17! . We then obtain E = IggA 2 . The kinetic energy is calculated 

 by integrating hgq 2 dxdydz, in which q is the total velocity. The limits of 

 the integration are the same as previously, and the additional limits are 

 z = to z = — oo. After dividing by j/U', we obtain E k = \ggA*. 



Per unit area we find again that the energy is half potential and half 

 kinetic and that the total energy is IgqA 2 , which is only half that of a two- 

 dimensional wave system of smaller amplitude (equation 11.14). 



