Question

How is a wave group formed ? Obtain an expression for the velocity of a wave group. What is the relation connecting the group and phase velocities ?

Answer 1

The velocity with which the overall envelope shape of the wave's amplitudes is called as the velocity of a wave group. The individual wavelets of differing wavelengths traveling at different speeds form a wave group.

We consider a wave packet as a function of position x and time t: α(x,t).

Let A(k) be its Fourier transform at time t = 0,

$$α(x,0)= \int_{ -∞}^ {∞} dk A(k)e^{ ikx } (1)$$

where ω is implicitly a function of k.

By the superposition principle, the wavepacket at any time t is

$$α(x,t)= \int_{ -∞}^ {∞} dk A(k)e ^{ i(kx-ωt) } (2)$$

where ω is implicitly a function of k.

After integrating (2) we got:

$$v_g = \frac {δω }{δk} (3)$$

where ω is the wave's angular frequency (usually expressed in radians per second), and k is the angular wavenumber.

Phase velocity vp and group velocity vg are related through Rayleigh's formula,

$$v_g=v_p(1-\frac {ω }{v_p} \frac {dv_p }{dω})^{-1} (4)$$