General Remarks on Waves 7 



Within a standing wave each individual water particle moves in an orbit 

 but, contrary to the progressive waves, the phase is the same for all particles, 

 and the amplitude of the movement is different for each particle. The points 

 where there is no vertical motion are called nodes, and the points where 

 there is no horizontal motion are called antinodes. We can study the pro- 

 perties of standing waves by superposing two progressive waves having the 

 same velocity and the same amplitude but opposite directions. From the 

 superposition results a simple standing wave. Figure 3 represents nineteen 

 different phases of such a case. In 1 both waves meet in the middle M of 

 the figure. In 2 both waves overlap for one-twelfth of the wave length. The 

 heavy line represents the sum of the ordinates of the two waves. In each 

 subsequent phase the original waves have progressed by one-twelfth of the 

 wave length. In 7 each wave has progressed by half its length. The superposi- 

 tion shows that both waves have cancelled each other out, because one wave 

 crest coincides with the wave trough of the other wave. At this particular 

 moment, all particles in the vertical direction go through their state of equi- 

 librium. Then, however, the vertical displacement starts again, in the middle 

 part downwards, in the sides upward. We can see that there are particles 

 which remain always at rest, because they are displaced by the progressive 

 wave going into one direction by exactly the same amount as by the other 

 progressive wave which travels in the opposite direction. Both waves cancel 

 each other and the particle remains at rest. These are the nodes N x , N 2 etc. 

 There are, furthermore, points called antinodes A x , A 2 etc. which have the 

 largest vertical displacements, up to the double amplitude of the progressive 

 wave. The antinodes, as well as the nodes, always remain in the same place. 

 The profile of the wave is subjected to continuous changes, but does not 

 move laterally and has no velocity, and therefore such waves have been called 

 standing waves. The wave length of a standing wave is equal to the distance 

 from one node to the next node, or from one antinode to the next antinode. 

 It can be shown that a standing wave results from a complete reflection of 

 a progressive wave on a vertical wall. The superposition of the incoming 

 wave and the reflected wave produces a standing oscillation, with always 

 an antinode at the wall. When r\ x is the vertical displacement on a wave moving 

 to the left, which hits a vertical wall a x = 0, and >/ 2 the vertical displacement 

 of the reflected wave moving in the oppostive direction, the superposition 

 of both waves will give a total vertical displacement of tj = % + ??.> . This gives 



r\ x = Asin((Jt — xx) , 



fj 2 = Asiniat-r-y.x) , (1.5) 



rj = rji+r) 2 = 2 A cos xx sin at . 



This standing wave has its nodes wherever cosy.x = 0, viz. at x = \{n + \)X 

 (n = 0, 1,2, ...). It has antinodes wherever cosx.v = ±1, viz. at 



x = \n'h . 



