The graph below shows that instantaneous displacements due to progressive iongitudinal wave travelling with a velocity `50ms^(-1)` along the direction ABCD. [Fig 14.4.8 (b)].
(a) What is the frequency of the wave ?
(b) What is the phase difference between the vibrations at C and D?
(c ) At which of the points A,B,C,D is the instantaneous particle velocity maximum forward?
(d) At which of the points, A,B,C,D is there a rarefraction?

Explanation

(a) `V=50ms^(-1), lambda=20cm=0.2m`
Frequency `=v=V/lambda=(50)/(0.2)=250Hz`
(b) Distance between C and D `=trianglex=5cm=0.05m`
Phase difference `triangle phi=(2pi)/(lambda) triangle x=(2pi)/(0.2) xx 0.05=pi/2`
(c ) Particle velocity `=(dy)/(dt)=-V (dy)/(dx)`
Particle velocity is maximum when `(dy)/(dx)` is maximum and negative. The point B has maximum ang negative value for `(dy)/(dx)`. This point corresponds to a compression.
(d) The value of `(dy)/(dx)` is positive at D. So at D excess pressure `p=-E. (dy)/(dx)` is positive. The point D corresponds to a rarefaction.