From the differentiation rule to a fingerprint
You have spent this rung turning derivatives into multiplication. The closing payoff is that you need not solve anything at all to learn how a Fourier transform equation moves its waves — you only have to probe it. Drop one single trial wave into the PDE and watch what the equation demands of it. The probe is e^(i (k x - omega t)): a pure wave with wavenumber k (how many wiggles per unit length) and angular frequency omega (how fast it oscillates in time). Substitute it, and every derivative collapses by the same rule you have been using all rung.
Here is the mechanism, slowed down. By the differentiation rule, an x-derivative on e^(i (k x - omega t)) just multiplies by i k, and a t-derivative multiplies by -i omega. So u_x becomes (i k) u, u_xx becomes (i k)^2 u = -k^2 u, and u_t becomes (-i omega) u. Because the wave is a common factor on every term, it cancels clean out, and what remains is a bare algebraic equation linking omega and k. That equation is the dispersion relation, written omega = omega(k). It carries no information about which wave you sent in; it is a property of the equation itself — its fingerprint.
Three fingerprints, side by side
Let us read three of this rung's equations off the same way. For the wave equation u_tt = c^2 u_xx, the trial wave gives (-i omega)^2 = c^2 (i k)^2, that is -omega^2 = -c^2 k^2, so omega = c k (taking the right-moving branch). For the Schrödinger equation i u_t = -u_xx, you get omega = k^2. And for the heat equation u_t = k u_xx — careful, two k's again, call the constant D — you get -i omega = D (i k)^2 = -D k^2, so omega = -i D k^2, a purely imaginary omega. That last one is the tell that heat does not oscillate at all; it decays. Three equations, three utterly different fingerprints.
- Wave equation, omega = c k: a straight line through the origin. omega is real, so it oscillates, and the line is straight, so it does not spread — non-dispersive.
- Schrödinger equation, omega = k^2: a curve, with omega real. It oscillates and it spreads, because different frequencies sit at different slopes — dispersive.
- Heat equation, omega = -i D k^2: omega is purely imaginary. Feeding it back into e^(-i omega t) gives e^(-D k^2 t), a decay, not an oscillation — no waves at all, just smoothing.
Now the word dispersion earns its keep. The wave equation's omega = c k is a perfectly straight line through the origin: every frequency travels at the identical speed c. The Schrödinger equation's omega = k^2 is a curve, so different frequencies travel at different speeds. "Dispersive" means exactly this — the relation between omega and k is not a straight line, so a wave built from many frequencies spreads out as its parts drift apart. The wave equation is the rare non-dispersive case, which is precisely why d'Alembert's formula lets a pulse travel rigidly as u(x,t) = F(x - c t) without changing shape.
Two speeds, and why they differ
Given omega(k), there are two natural speeds to read off, and conflating them is the classic beginner's mistake. The first is the phase velocity, v_p = omega / k: how fast a single crest of one pure wave slides along. It is found by asking how fast the point where the phase k x - omega t stays constant must move; setting the phase fixed gives x = (omega / k) t. This is the speed of an individual ripple, the thing your eye locks onto when you watch one wave.
But a real signal is never a single pure wave — it is a packet, a bundle of nearby frequencies added together, with an envelope that bulges where the waves agree and cancels where they clash. The envelope, the lump that actually carries the signal and its energy, moves at a different speed: the group velocity, v_g = d(omega)/dk, the slope of the dispersion curve. The reasoning is a one-line stationary-phase argument: among all the frequencies in the packet, the place where they reinforce rather than cancel is where their phases change in step, and that is where d(omega)/dk sets the pace. So the crest speed is the ratio omega/k; the packet speed is the slope d(omega)/dk.
phase velocity v_p = omega / k (a ratio -- one crest) group velocity v_g = d(omega)/dk (a slope -- the packet & its energy) wave equation omega = c k -> v_p = c, v_g = c (equal!) Schrodinger omega = k^2 -> v_p = k, v_g = 2k (v_g = 2 v_p) deep water omega = sqrt(g k) -> v_p = sqrt(g/k), v_g = v_p / 2
Why a packet spreads — the stationary-phase view
There is a deeper reason a dispersive packet not only moves but slowly spreads, and it sits right inside the inversion integral from this rung. After transforming, the solution is u(x,t) = (1/2pi) integral of U_0(k) e^(i (k x - omega(k) t)) dk — every frequency present in the initial data, each spun forward by its own e^(-i omega(k) t). To recover where the packet is, you ask which frequencies add up rather than cancel, and that is the method of stationary phase: for large t the integral is dominated by the k where the phase k x - omega(k) t is momentarily flat, i.e. where x = omega'(k) t. That is the group velocity again, now earned from the integral itself.
And the spreading? Because v_g = omega'(k) itself depends on k whenever omega(k) is curved, different parts of the packet arrive at different group velocities, so the lump stretches and flattens as it travels. This is the genuine, honest reason a quantum wave packet or a water-wave group broadens over time — not energy loss, but pure geometry of the dispersion curve. Contrast the heat equation, whose imaginary omega = -i D k^2 means the high frequencies do not race ahead but simply die, so its profiles smooth rather than disperse. Dispersion and dissipation are different fates, and the dispersion relation tells you, at a glance, which one an equation suffers.
Honest limits, and where it breaks
Be clear about what this fingerprint can and cannot do. It is exact only for linear, constant-coefficient equations on the whole line — exactly the world this rung built. There a single wave is a genuine solution, the travelling-wave e^(i(kx - omega t)) survives intact, and superposition lets you sum waves to make any initial data. Step outside that world and the dispersion relation either fails to exist or becomes only a local, approximate guide.
Nonlinearity is where the story turns subtle and beautiful. In the nonlinear Schrödinger equation or the sine-Gordon equation, the linear part still disperses — its omega(k) is curved and wants to spread any packet — but a nonlinear term pulls inward and refocuses it. When the two balance exactly, you get a soliton: a lump that travels without spreading at all, a wave that defies dispersion by the precise grace of nonlinearity. So a dispersion relation read off the linearized equation is the start of the story for these models, not the end; it tells you what the wave would do if left alone, and the nonlinearity tells you what saves it.
One last caution, the same one that opened this rung. A pure wave e^(i(kx - omega t)) is not a square-integrable function — it has infinite total energy and is not, by itself, a legal element of the spaces where the inverse transform lives. It is a probe, an idealised building block; real, physical solutions are packets, integrals of these probes weighted by the data. The dispersion relation is none the worse for that. It is the cleanest example in the whole rung of the rung's own motto: transform, and the physics you want is sitting there in algebra, waiting to be read.