72 Generation, Growth and Propagation of Waves 



conditions, will explain the transient state in wave height when the wave 

 advances into calm areas. 



The process of the transmission of energy, according to observations on 

 wave propagation in deep water was qualitatively expressed in the following 

 form by Gaillard (1935, p. 194) and quoted here from Sverdrup and 

 Munk (1942): 



"Suppose that in a very long trough containing water originally at rest, 

 a plunger at one end is suddenly set into harmonic motion and starts generat- 

 ing waves by periodically imparting an energy \E to the water. After a time 

 interval of n periods, there are n waves present and equals the time in periods 

 since the first wave entered the area of calm. 



"If the position of a particular wave within this group is indicated by m 

 and equals the distance from the plunger expressed in wave lengths, m = 1 

 represents the wave just generated by the plunger, m = \{n-\-\) will be the 

 centre wave and m = n the wave which has travelled farthest. Let the waves 

 travel with constant velocity c and neglect friction." 



The first wave generated by the plunger's first stroke will have the energy \E. 

 One period later this wave will have advanced one wave length, leaving 

 behind one-half of its energy or IE; it now occupies the space of a wave length 

 of the previously undisturbed area, into which it brings the energy \E. A second 

 wave has been generated by the plunger occupying the position next to the 

 plunger, where \E was left behind by the first wave. The energy of the second 

 wave equals %E+ \E — \E. This is repeated, and when three waves are present 

 on the water surface, the one which has just advanced into the undisturbed 

 area has an energy of IE, the second one of this series \E-\- | of IE = f £, 

 the third one I of %E+%E = IE, and so forth. Table 11 shows the dis- 

 tribution of energy in such a short wave train. 



In any series, n, the deviation of the energy from the value \E is sym- 

 metrical about the centre wave. Relative to the centre wave all waves nearer 

 the plunger show an excess of energy and all waves beyond the centre wave 

 show a deficit. 



For any two waves at equal distances from the centre wave the excess 

 equals the deficiency. 



In every series, n, the energy first decreases slowly with increasing distance 

 from the plunger, but in the vicinity of the centre wave it decreases rapidly. 

 Thus, there develops an "energy front" which advances with the speed of 

 the central part of the wave system, that is, with half the wave velocity. 



According to the last line in Table 11 a definite pattern develops after 

 a few strokes: the wave closest to the plunger has an energy E(2 n — 1)/2" which 

 approaches the full amount E with increasing 77, the centre wave has an 

 energy \E, and the wave which has travelled the greatest distance has very 

 little energy (E/2 n ). 



An exact distribution of energy and therefore also of the wave height 



