The simplest PDE of all describes something being carried along at constant speed — a dye drifting down a pipe, a signal moving along a wire. With u(x, t) the amount at position x and time t, the transport equation (or advection equation) is

It says the time-change of u is exactly balanced by its spatial slope, scaled by a speed c. First order, linear, homogeneous — and yet it already contains the essential idea of propagation.

The solution is the initial shape, sliding

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

Check it. By the chain rule, u_t = -c\,f'(x - ct) and u_x = f'(x - ct), so u_t + c\,u_x = -c f' + c f' = 0. ✓ The whole profile simply translates to the right at speed c, rigid and undistorted — no spreading, no decay. The graph at time t is the starting graph shifted by ct.

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.