General Remarks on Waves 



13 



occupied in the group is now occupied in succession by other waves which 

 have moved forward from the rear. It appears as if, on the front end of 

 a group, the waves constantly disappear, whereas new waves appear at the 

 rear. The group is then continuously composed of waves which travel through 

 them. The behaviour of such groups can be best explained, according to 

 Lord Raleigh (Lamb, 1932, p. 382), by the superposition of two systems 

 of waves of equal amplitude, but of a somewhat different wave length and 

 velocity. This overlapping will give the following equation of the free surface: 



)) = As'm(xx — <Jt)—As'm(x'.\—<7't) 



= 2A cos 



x — x a — a 



sin 



x^x a -\- a' 

 — s — x ~ — t 



(1. 10) 



As x and x' differ only very slightly x = x' -{- Ax, the cosine in the first 

 part of this expression changes its value very slowly with x; so that, at any 

 instance, the wave profile has the form of a sine curve, in which the 

 amplitudes vary between the values and 2 A. The surface thus has the 

 appearance of a succession of groups of waves separated, at equal intervals, 

 by strips of almost smooth water. Figure 6 shows an example of the over- 

 lapping of two such waves: r\ = 1000cos6.\-sin60.v, for which x = 66° and 



Fig. 6. Superposition of two waves with slightly different wave lengths (ratio 9 : 11). Group 



maxima 30 units apart. 



*' = 54°, and which correspond to the wave lengths of approximately 5 • 45 c 

 and 6 • 66°. The amplitude becomes = when 



6.Y = 



in intervals of 30 



(2/7+ l)ijr (n = 0, 1 , 2...) , this means, 



It looks as if the entire train of waves is composed of 

 wave groups, each of which has a length of 30°. The periodical phenomenon 

 under consideration can be explained by a superposition of two waves having 

 approximately the same wave length. It is easy to follow the path of such 

 wave groups by means of Fig. 7. It shows the superposition of two wave 

 systems whose velocities c x and c 2 correspond to a ratio of 17 to 15, and 

 their wave length to the ratio 5 to 4. One can easily see that the wave groups 



