51 Wave dispersion and group velocity

Wave components need not travel at the same phase speed (i.e., the speed at which crests and troughs travel).

If they do have the same phase speed, the waves are called nondispersive. In this case, the interference pattern (the sea surface shape created by their sum) will not change over time, but will simply translate through space at the phase speed. For forcing that occurs over a short amount of time, this moving shape is called a “pulse”.

If wave components do not have the same phase speed, then the waves are called dispersive. In this case, a pulse will break up. Very long waves and very short waves will form low amplitude trailing edges, and the bulk of the energy from the generation event will travel at a speed called the group speed. For an animation of dispersive and nondispersive behavior, see Dan Russell’s webpage Waves in a Dispersive Medium.

Key Takeaways

For dispersive waves, wave energy travels at a group speed that is distinct from the speed of wave component crests (their phase speed).

For nondispersive waves, wave energy travels at the phase speed (which is the same for all wave components).

As the waves from a storm become very spread out in space, a relatively narrow range of frequencies will be found at a given place, and this set of waves will be associated with a particular group speed. We can look at this in the limit of two similar frequencies of traveling waves that will “beat” in a similar way as we saw for standing oscillations, but here we will add cosines that represent traveling waves.

    \[ \eta(x,t) = A \hspace{2pt} cos(k_1 x-\omega_1 t)+A \hspace{2pt} cos(k_2 x-\omega_2 t) \]

Let’s define the average and differences of the wavenumber and frequencies

    \[ \overline{k}=\frac{k_1+k_2}{2} \quad \overline{\omega}=\frac{\omega_1+\omega_2}{2} \]

    \[ \Delta k = k_1-k_2 \quad \Delta \omega = \omega_1-\omega_2 \]

Then, using the same trignometric identity as for standing wave superposition, we find

    \[ \eta(x,t)=2Acos\left( \frac{\Delta k}{2}x- \frac{\Delta \omega}{2}t\right) cos\left( \overline{k}x-\overline{\omega}t \right). \]

The interference pattern travels at a lower frequency, \Delta \omega / \Delta k, as animated in the final image on Dan Russell’s webpage Superposition of Waves.

For a continuous spectrum of waves, we identify the finite difference expression with a continuous derivative, \partial \omega / \partial k. This assumes a function that relates a specific wavenumber to a specific frequency, \omega(k). Such a function, derived from the physics of different kinds of waves, is called the dispersion relationship.

Key Takeaways

The two important wave speeds that can be determined from the dispersion relationship, \omega(k), are

The phase speed, representing the speed of travel of the crests of individual wave components

    \[ C_P = \frac{\omega}{k} \]

The group speed, representing the speed of travel of wave energy

    \[ C_G = \frac{\partial \omega}{\partial k} \]