Mathematical Methods for Physicists: A concise introduction - Site Map

WAVE PACKETS AND GROUP VELOCITY
Figure 4.18.
Superposition of two waves.
The modulation of the wave is repeated inde®nitely in the case of superposition of
two almost equal waves. We now use the Fourier technique to demonstrate that
any isolated packet of oscillatory disturbance of frequency ! can be described in
terms of a combination of in®nite trains of frequencies distributed around !. Let
us ®rst superpose a system of n waves
…x; t† ˆXn
jˆ1
A j e i…k jx! j t† ;
where A j denotes the amplitudes of the individual waves. As n approaches in®nity,
the frequencies become continuously distributed. Thus we can replace the summation
with an integration, and obtain
…x; t† ˆ
Z 1
1
A…k†e i…kx!t† dk;
…4:34†
the amplitude A…k† is often called the distribution function of the wave. For
…x; t† to represent a wave packet traveling with a characteristic group velocity,
it is necessary that the range of propagation vectors included in the superposition
be fairly small. Thus, we assume that the amplitude A…k† 6ˆ0 only for a small
range of values about a particular k 0 of k:
A…k† 6ˆ0; k 0 "