70 Generation, Growth and Propagation of Waves 



nature. The same follows from (IV. 2). Nodes appear when w = {2n — \)hn-\-nl4 

 (n = 1,2,3, ...). The first one at the time t x is near x x , the next one near x 2 ; 

 the wave length then is x z — x x and 



4x ± 4x 2 



If the difference between both quantities w remains equal, then at a later 

 time to, the quantities gt 2 , x x and x 2 and also the distance x 2 — x x have in- 

 creased correspondingly. Each wave, therefore, lengthens while propagating, 

 and this explains the accelerating pace of the wave disturbance. For suf- 

 ficiently great values of t, the changes in length and the height from wave 

 to wave are very gradual, so that in first approximation a considerable number 

 of consecutive waves may be represented by a curve of sines. In a progressive 

 wave for this short distance at— xx = (gt 2 )/(4x). When we vary in this ex- 

 pression only the time /, we obtain the wave period 



T = — . (IV.3) 



gt K 



Whilst if we vary x alone we obtain the wave length 



Stix 2 



gt* 



The wave velocity then is 



(IV. 4) 



This, however, is the velocity of propagation of an infinitely long train simple- 

 harmonic surface waves according to Stokes equation (11.11). Each individual 

 wave, therefore, behaves in first approximation like other simple water waves. 

 But when we consider a group of waves having approximately a wave length 

 A , then at a time t and at a locality x , h = Snxl/gtl and the velocity of 

 the group 



$-W& (lv - 6) 



which is half that of the component wave. The group does not maintain 

 a constant amplitude as it advances. This is in agreement with the relation 

 on p. 12. 



These theoretical results are confirmed by many observations, in wave 

 tanks and in nature. Only a portion of the energy of a wave advancing 

 in calm water travels along with the wave form, whereas the remaining portion 

 of this energy remains available for the next wave. In this manner, each single 

 wave in a wave train in deep water acquires energy from the preceding wave 

 and leaves a portion of its energy for the wave following it. Due to this 



