20 



Theory of Short and Long Waves 



trough is u 2 = c+ 2nrJ.T. The wave height is h = 2r, so that according to the equation of Bernoulli 

 for steady currents 



Wg— u\ = 2^/? = 4gr , 



on account of the equality of pressure. With these values for w t and u 2 the left side of the equation 

 becomes Sjicr/T, and c = gT(2n, which corresponds to (11.11). 



Table 1 . Characteristic values of surface waves 



Ratio: 



Ratio: 



Ratio: water velocity 



The streamlines of the wave motion in surface waves, relative to a reference 

 system at rest, are shown in Fig. 10. The broken-line circles represent the 

 orbits for individual particles. It can be seen how rapidly the dimensions of 

 these orbits decrease with increasing depth (see vol. I, pt. 2,). In a depth 

 of one wave length the radius is reduced to Ae~' lTX * i. e. approximately the 

 one five-hundredth of its size at the surface. 



The energy of a wave system of progressive waves can easily be calculated. 

 A narrow strip of unit width and of unit length dx, parallel to the direction 

 of propagation, with a vertical elevation at the surface r\ according to (II. 8), 

 has a potential energy Igorfdx. In the unit area the potential energy amounts to 



