After the first complete stroke one wave will be 

 present and its energy is E/2. One period later this 

 wave has advanced one wave length but has left one- 

 half of its energy or E/4 behind. It now occupies a 

 previously undisturbed area to which it has brought 

 energy E/4. In the meantime, a second wave has been 

 generated, occupying the position next to the plunger 

 where E/4 was left behind by the first wave. The 

 energy of this second wave equals E/4 + E/2 = 3E/4. 

 Repeated applications of this reasoning lead to the 

 results shown in Table 2-1. 



The series number n gives the total number of 

 waves present and equals the time in periods since 

 the first wave entered the area of calm; the wave 

 number m gives the position of the wave measured from 

 the plunger and equals the distance from the plunger 

 expressed in wave lengths. In any series, n, the 

 deviation of the energy from the value E/2 is 

 symmetrical about the center wave. Relative to the 

 center wave all waves nearer the plunger show an 

 excess of energy and all waves beyond the center wave 

 show a deficit. For any two waves at equal distances 

 from the center 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 center 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 2-1 a definite 

 pattern develops after a few strokes: the wave closest 

 to the plunger has an energy E(2"-l)/2" which approaches 

 the full amount E, the center wave has an energy E/2, 

 and the wave which has traveled the greatest distance 

 has very little energy (E/2^) . 



Table 2-1. Distribution of Wave Heights in a Short Train of Waves 



2-32 



