Answer to Question #164485 in Physics for Mst. Ohida Khatun

Question

(a) Discuss the phase velocity and group velocity in details with net diagram.  (b) A particle execute simple harmonic motion about the point x = 0. At t= 0, it has displaced y = 0.37 cm, and zero velocity. If the frequency of motion is 0.25/s. Calculate the period, amplitude, the maximum speed and the maximum acceleration

Answer 1

a) In order to convey information, something more than a simple harmonic wave is needed. However, the superposition of many such waves of varying frequencies can result in an "envelope" wave and a carrier wave within the envelope. The envelope can transmit data. A simple example is the superposition of two harmonic waves with frequencies that are very close and of the same amplitude.

The envelope (the green line) is given by u₁ and travels at the group velocity. The carrier wave (the blue line) travels at the phase velocity and is given by u₂. The wave packet moves at the group velocity. It is the envelope which carries information. Group velocity and phase velocity are not necessarily the same. 

b)

$$T=\frac{1}{0.25}=4\ s\\A=0.37\ cm\\V=0.37(2\pi)(0.25)=0.58\frac{cm}{s}\\a=0.37(2\pi)^2(0.25)^2=0.81\frac{cm}{s^2}$$