Question

Sinusoidal waves are generated on an infinitely long rope. One wave, y1(x,t), moves to the left and has amplitude y0, wave number k1, and angular frequency ω1; the other, y2(x,t), moves to the right with amplitude y0, wave number k2, and angular frequency ω2. Each wave has the same phase. (a) Express yi(x,t) for each of the waves. (b) Assuming that the rope is a nondispersive medium, what is ω2 in terms of k1, k2, and ω1? In parts (c) and (d), assume that k2=k1+δk and ω2=ω1+δω. Take the speed of traveling waves on the rope to be v. (c) What is δω in terms of δk and v? (d) Superpose these waves, assuming that δk and δω are small. Use the notation kav = (k1+k2)/2 and ωav = (ω1+ω2)/2. Show that the result of the superposition is two waves, one moving at the low speed vl = vδk/2kav and wavelength 2π/kav, and one moving at the high speed vr = 2vkav/δk and the long wavelength 4π/δk. (20%)

Answer 1

a)
  
  
So,
  
b)
  In a non-dispersive medium, velocity of wave is independent of frequency. So,
  

  
so,
   
   
  
  

d)
  
  
  
  
So, this superposed wave is product of two waves (i.e., of the for (x-vt) form. ).
For the 1st part
  
  the wave speed is
  
And the wavelength is (wavelength is inverse of the wavevector, and for the 1st part, the wave vector is k_{av}).
  
And for the 2nd part,
   
   the wave speed is
  
where, we have used the relation that
   
And the wavelength is (wavelength is inverse of the wavevector, and for the 1st part, the wave vector is \delta k/2).