Wave Packet - Basic Behaviors of Wave Packets

Basic Behaviors of Wave Packets

As an example of propagation without dispersion, consider wave solutions to the following wave equation:

where c is the speed of the wave's propagation in a given medium. Using the physics time convention, exp(−iωt), the wave equation has plane-wave solutions

where

This relation between ω and k should be valid so that the plane wave is a solution to the wave equation. It is called a dispersion relation.

To simplify, consider only waves propagating in one dimension (extension to three dimensions is straightforward). Then the general solution is

in which we may take ω= kc . The first term represents a wave propagating in the positive x-direction since it is a function of x−ct only; the second term, being a function of x+ct, represents a wave propagating in the negative x-direction.

A wave packet is a localized disturbance that results from the sum of many different wave forms. If the packet is strongly localized, more frequencies are needed to allow the constructive superposition in the region of localization and destructive superposition outside the region. From the basic solutions in one dimension, a general form of a wave packet can be expressed as

.

As in the plane-wave case the wave packet travels to the right for ω(k)=kc (since then u(x,t)=F(x−ct)) and to the left for ω(k)=−kc (since then u(x,t) = F(x+ct)).

The factor comes from Fourier transform conventions. The amplitude A(k) contains the coefficients of the linear superposition of the plane-wave solutions. These coefficients can in turn be expressed as a function of u(x,t) evaluated at t=0 by inverting the Fourier transform relation above:

.

For instance, choosing

we obtain

and finally

The nondispersive propagation of the real or imaginary part of this wave packet is presented in the above animation.

As an example of propagation with dispersion, consider solutions to the Schrödinger equation (with m and ħ set equal to one)

yielding as dispersion relation

Once again, restricting ourselves to one dimension the solution to the Schrödinger equation satisfying the initial condition is found according to

An impression of the dispersive behaviour of this wave packet is obtained by looking at

(note that |u(x,t)| itself is not a solution of the Schrödinger equation).

It is evident that this dispersive wave packet, while moving with constant group velocity k₀, has a width increasing with time as .