WAVE PACKETS

When we superpose i.e. sum two monochromatic waves with nearly equal frequencies. Of course, we can have a group of many waves having different frequencies and in most physical situations this is usually the case. The different frequencies may be discrete or they may cover a continuous range. (We are familiar with the concept of a continuous frequency distribution in the case of white light that contains a continuous range of frequencies from blue to red light.) Figure 1(a) illustrates an important example of a continuous frequency distribution that occurs in many physical situations. This distribution lies symmetrically about a central frequency ωo and has a width

Figure 1: (a) An important example of a continuous frequency distribution that occurs in many physical situations. This distribution lies smoothly and symmetrically about a central frequency ω_o. The width Δω of the distribution is small compared with ω_o. (b) The wave packet, of temporal width Δt, resulting from the superposition of the frequency components of the distribution in (a).

......(1)
ω that is small compared with ω_o. It also has a smooth profile. The result of
superposing the frequency components of this distribution is shown on a time axis
in Figure 1(b). We obtain a pulse of waves or wave packet that is highly localised
in time with a width Δt. The wave packet travels at the group velocity which is
given by the same equation (1), v_g = dω/dk, that we had for the case of just
two monochromatic waves. The energy is concentrated around the amplitude maximum
and travels at the group velocity as does any information carried by the wave
packet. We will show that the width Δω of the frequency distribution
and the temporal width Δt of the wave packet are related by ΔtΔω ≈ 2π. This
is called the bandwidth theorem. This is a very important and general result that
applies to a wide range of physical phenomena where there is a disturbance ψ(t)
that is localised in time, i.e. some sort of wave pulse. This relationship between
t and ω does not depend on the specific shape of ψ(t) so long as it has the
characteristic that defines a pulse, i.e. that ψ(t) is different from zero only over
the limited time interval Δt. It follows that to obtain pulses of shorter duration Δt,
we have to increase the range of frequencies Δω.
There are many examples of wave pulses and packets in physical situations. For
example, narrow pulses of light are passed down optical fibres for communication
purposes. Higher data transmission rates require pulses of very short duration Δt.
Consequently, the sending and receiving equipment needs to operate over correspondingly
high frequency bandwidths. On the research side, scientists are making
wave packets of light that contain just a few cycles of optical oscillation, corresponding
to pulse lengths of femtoseconds (∼10⁻¹⁵ s). Wave packets also have