Suppose you made observations of water waves, say standing out on the pier with a stopwatch and pacing off a few lengths. You would collect data that consisted of pairs, wavenumber and frequency. You would have to go at different times to see different wave trains. These data points would not fill all of (k, \omega) space, but would fall along a line given by the dispersion relation

\omega^2 = k.

Here is an alternate way to plot the data. Instead of the point (k, \omega) plot the line in (x,t) space given by

k x + \omega t = 2 \pi.

Instead of a line passing through a set of data points, we can now draw the envelope of these data lines. This will contain just as much information as the dispersion relation, and have some visualization advantages.

If we drew all of the lines

k x + \omega t = 2 \pi n

for integer values of n, then we would have a spacetime diagram for the entire wavetrain. This can be constructed from the single line tangent to the envelope knowing that the lines are all parallel, and that one must pass through the origin.

In addition, we will prove later that the line from the origin to the point of tangency with the envelope is a characteristic line (in the sense of PDEs), which represent the motion of energy and information with a speed given by the group velocity. For water waves the group velocity is half the speed of the wave crests (phase velocity).

The advantage of this representation of water waves is that is can be done in the same wave in higher dimensions. The group velocity will always define a characteristic line in spacetime, and will be represented by a vector. The motion of the wavecrests will be given instead of lines, by hyperplanes. Only in two dimensions can the wavecrests be assigned a (misleading) velocity. In higher dimensions, their motion must be represented by a 1-form, iconized by the pair of parallel hyperplanes.