4 General Remarks on Waves 



Equation (1.3) shows that the waves advance unchanged in the -\-x di- 

 rection with a velocity c = x/t = l\T . c is called the wave velocity and 

 l/T = v is called the frequency of the wave. Often the quantity 2njT = a 

 is used which is called the "angular velocity" of the wave. We can similarly 



7 8 9 10 II 12 13 W 15 



Fig. 1. Development of an harmonic progressive transverse wave. 



introduce a quantity 2n\l = x and call it the "wave number", because it 

 indicates the number of waves over a distance 2.-r. We thus obtain the fun- 

 damental equations applicable to all waves: 



X = cT, 



a — xc 



7] = A sin(o7 — xx) 



(1.4) 



In the wave motion discussed so far each particle of the medium executed 

 movements only perpendicular to the direction of propagation of the wave. 

 In water waves we have to add a rhythmic motion in the direction of pro- 

 pagation of the wave, so that each particle, with the passage of a wave, 

 executes an orbit. In this way a system of waves moving with a definite 

 velocity is developed on the surface which is called a wave train. 



In case the particles move in circular orbits, we have to replace Fig. 1 

 by Fig. 2. The water particles numbered from to 15 move in their orbits 

 with the same uniform speed in such a manner that particle 12 starts its 

 movement at the exact moment when particle has completed its circle. 

 Therefore, it is in the same phase as particle and, consequently, the distance 

 to 12 represents the wave length. In this particular case the wave profile 

 is no longer harmonic, because the wave crest and the wave trough have 

 a different form, inasmuch as the wave crest is shorter and steeper, whereas 

 the wave trough is wider and flatter. This curve is a trochoid (p. 28). Within 

 a progressive wave the horizontal flow at the wave crest is in the direction of 



