General Remarks on Waves 3 



Let us now suppose that the individual water particles of the surface 

 execute such harmonic oscillations in a vertical plane and that the energy 

 is transmitted from one particle to another successively in horizontal direction, 

 with each particle starting its motion somewhat later, but this difference in 

 time between each particle remaining the same. Let us designate this time 

 interval as t. Consider n successive particles; particle will execute the 

 harmonic movement as expressed by (LI), particle 1 the motion 



. . 2n . 

 )h = Asm ~ (f-r), 



and in general the motion of particle i will be expressed by 



, . 2n . 

 i]i = A sin — (/ — it) . 



Particle n should start its harmonic oscillation at the very moment when 

 particle has completed its first harmonic oscillation, then m = T. We can 

 replace t by Tin and the motion of particle i is given by 



r}^ A sin 2 ^ It- ~t\. (1.2) 



If we designate the horizontal distance between particle and particle n 

 as X, the distance between particle and particle i as x, then x: l = i: n 

 and we obtain from (1.2) the equation for the motion of each particle in 

 the following form 



r} = A sin 



In In 



(1.3) 



This is the general equation for a progressive wave travelling in the di- 

 rection +jc and of an harmonic type. Figure 1 illustrates the development of 

 such a wave for 15 consecutive equidistant particles. Particle has com- 

 pleted its harmonic oscillation at the exact moment of the start of that of 

 particle 12. This moment corresponds to the XII row of Fig. 1. A is called 

 the wave length; it is the horizontal distance between two particles which 

 are in the same phase of motion; it is, furthermore, the horizontal distance 

 between two corresponding points — like from trough to trough or from 

 crest to crest — for a given time. The form of the wave is an harmonic func- 

 tion and is given by the equation: 



rj = A sin I C y x ) 



The greatest elevations are called wave crests, the greatest depression 

 wave troughs; their vertical distance is the wave height, whereas the wave 

 amplitude is one-half of the wave height. 



