A factory upstream spills a little dye into a river. The river doesn't stir the dye into nothing, and it doesn't leave it sitting still — it simply carries the whole blob downstream at the river's flow speed, keeping its shape almost exactly as it goes. Pluck a rope and send a single bump along it: the bump doesn't spread out or fade — it glides down the rope, unchanged, at a fixed speed. Both are the same physical idea: something being transported rather than diffusing or oscillating.

With u(x, t) the amount of "stuff" (dye concentration, rope displacement, anything) at position x and time t, the transport equation (also called the advection equation) is

It says the time-change of u is exactly balanced by its spatial slope, scaled by a constant speed c. It is the simplest PDE you can write down — first order, linear, homogeneous, one term of each kind — and yet it already contains the essential idea of propagation that every wave equation in this course builds on.

The solution is the initial shape, sliding

Claim: if the profile at t = 0 is u(x, 0) = f(x), then for all later time

Check it. By the chain rule, differentiating f(x - ct) with respect to t brings down a factor of -c (the inner derivative of x - ct with respect to t), while differentiating with respect to x brings down a factor of 1:

Substitute both into the equation: u_t + c\,u_x = -c f'(x-ct) + c f'(x-ct) = 0. ✓ It works for any differentiable f — the whole profile simply translates to the right at speed c, rigid and undistorted: no spreading, no decay, no ringing. The graph at time t is exactly the starting graph, shifted sideways by ct.

It's linear too — so solutions still add up

Notice that u_t + c\,u_x = 0 is exactly the shape of equation the superposition principle applies to: it's linear (only first powers of u and its derivatives appear) and homogeneous (no term is free of u). That means if f_1 and f_2 are two different starting profiles, each carried along on its own as f_1(x-ct) and f_2(x-ct), then dumping both profiles into the river at once gives exactly

two independent bumps sliding along side by side at the same speed, each utterly unbothered by the other's presence. Two pollutant plumes released from different pipes into the same river drift downstream without ever interacting — which is a much less obvious fact for, say, two waves crossing on the surface of the ocean, where nonlinear effects can make them nudge each other.

Worked example: tracking a bump by hand

Suppose the initial dye concentration along a river (measured in kilometres from a reference point) is a narrow bump centred at x = 0, say f(x) = e^{-x^2}, and the river flows at c = 3 km/h. Where is the peak of the dye cloud after t = 2 hours?

The formula does all the work: u(x, t) = f(x - 3t), and f peaks exactly where its argument is 0, i.e. at x - 3t = 0, so x = 3t. At t = 2, that's x = 3 \times 2 = 6 km downstream. Not just the peak, either — the entire bump, spikes and tails and all, has simply slid 6 km to the right without changing width or height at all. If you wanted the concentration at some fixed sampling station 4 km downstream at that same moment, you'd read it off as u(4, 2) = f(4 - 6) = f(-2) = e^{-4} \approx 0.018 — the leading edge of the bump has only just started to arrive there.

Watch it travel

A Gaussian bump f(x) = e^{-x^2} carried by u(x, t) = e^{-(x - ct)^2}. Advance the time t and the bump glides along, perfectly preserving its shape; change the speed c (even make it negative) to send it the other way. The faint curve marks the original position at t = 0.

The paths where nothing changes

There's a second way to see the same fact that turns out to be the seed of a much more general method. Look at the family of straight lines in the (x,t)-plane defined by x - ct = \text{constant}. Along any one of these lines, the quantity x - ct never changes — and since u(x,t) = f(x-ct) depends on (x,t) only through that combination, u is perfectly constant all along each such line. These special paths are called characteristic lines: riding along one at exactly speed c, you'd see the dye concentration around you never change, even though it's changing at every fixed location on the riverbank.

Take c = 3 again. The characteristic through (x,t)=(0,0) is x - 3t = 0, i.e. x = 3t — precisely the path of the peak we tracked by hand above. A boat drifting exactly at the river's flow speed, starting at the peak, would sit at the crest of the dye bump forever; a boat anchored to the bank (fixed x) would instead watch the concentration rise sharply as the bump arrives and fall again as it passes — two completely different experiences of the very same solution, depending only on which path through the (x,t)-plane you follow.

This is more than a cute reformulation. For the plain transport equation, the characteristics are just parallel straight lines, all with the same slope — but the same idea (find the paths along which the PDE reduces to "nothing changes," then follow them) turns out to solve a whole family of harder first-order PDEs, even nonlinear ones with curving characteristics. That general technique is the method of characteristics, and this equation is its simplest possible example.