14 



General Remarks on Waves 



Fig. 7. Behaviour of wave groups resulting from the superposition of two wave systems 

 (ratio of wave lengths 5:4); (ratio of wave velocities 17: 15) (From Grimsel-Tomaschek, 

 Lehrbuch der Physik, vol. 1, p. 405). 



travel much slower than the separate waves. The behaviour of the separate 

 waves within the groups can be seen in this figure. The distance between 

 the centres of two successive groups, according to (1. 10) is 2n/(x — x') = 2n\Ax 

 and the time needed by the system to cover this distance is 2ji/(<j—o') = 2n\Ao. 

 The velocity of group C is consequently (a— a')/(x — x') = Aa/Ax, or in case 

 there is only a slight difference between the respective values of the de- 

 nominator and the numerator, C = da/dx. With A = 2n/x, one obtains 

 with (1.4) C = dxc/dx or C = c—l{dcjdX) (1. 11) when c is the wave velocity. 

 Generally the wave velocity increases with the wave length, and therefore 

 the group velocity is smaller than the wave velocity. Progressive waves at 

 the surface of a water layer of a depth h have a wave velocity of 

 c = [g/^tanhx/?] 1 ' 2 (see p. 18, equation 11.10). The group velocity of a train 

 of such waves will be 



C = \c 



1 + 



2xh 



sinh2x/i 



(1.12) 



