We will now re-introduce the time dependence for real traveling waves. Below you will see one of the localized wave packets that we studied in Section 2c. It is found by adding ±0.05,0.1,0.15,0.2*k to the fundamental. Each component has an amplitude of 1.

Dispersion is the manner in which the frequency and the wave number are related in a medium.

Light waves have a linear dispersion in a vacuum but in water they have a
nonlinear dispersion. This nonlinear dispersion can cause a white light
spectrum to be changed into a rainbow. For light in a vacuum, every frequency travels at the
same speed, c. E = pc = ħkc
= ħw
. w
= kc. The index of refraction of visible light in glass has a component
that varies as a k^{2}, to a first approximation. Thus, visible light
in glass has an inverse dispersion relation. w
= w(1/k)

The probability density function, Psi(x,t), for massive particles has a nonlinear, quadratic dispersion. E
= ħw
= p^{2}/2m. w
= ħk^{2}/2m.

**Student Exercises:**

- Describe the motion of the traveling wave in a medium with a linear dispersion.
- Describe the motion of the traveling wave in a medium with a nonlinear, quadratic dispersion. Notice that at t = 0 the starting wave profile is the same as for the linear dispersion. Look at t = 10. Why do you think the packets appear to widen with time?
- For the quadratic case you will notice that as time progresses, the packets first seem to disappear and then the waves rephase to form other combinations of packets. For example, at t » 100, there appear to be five well defined packets. Why? At what time does the original configuration rephase? Be patient.

Each point along the x-axis oscillates in a complicated (non-sinusoidal)
manner. Before we complete this Section, write the time dependent function
for the **linear** dispersion wave used in this Section. Now go to the
remaining part by clicking on Initialize.