In the method of characteristics, every family of characteristics we met was well-behaved: the curves were fixed by where they started, and no two of them ever touched. That good behaviour had a hidden assumption baked into it — the speed a characteristic travels at never depended on the very quantity u it was carrying. Break that assumption, and the whole tidy picture can fall apart in a genuinely new way: fast-moving parts of the solution can catch up to slow-moving parts ahead of them, their characteristics collide, and a sharp jump — a shock — erupts out of a perfectly smooth starting condition.

Equations where the speed depends on u itself are called quasilinear; the famous toy example is the inviscid Burgers' equation

a model of a fluid where each point simply moves at a speed equal to its own height. Following the same logic as before, along curves with \frac{dx}{dt} = u we still get \frac{du}{dt} = 0 — u is still constant along a characteristic. The twist is that each characteristic now carries its own private speed u(x_0, 0), so the lines are no longer parallel, and nothing stops them crossing.

The mechanism: a traffic jam in miniature

The cleanest way to feel why this happens is to picture cars on a motorway instead of a fluid. Traffic-density models are built on exactly this kind of quasilinear PDE, with one simple physical rule: denser traffic moves slower. A car in an open, sparse stretch of road can cruise; a car packed into a dense stretch has to crawl.

Now imagine a region of fast, sparse traffic sitting just behind a region of dense, slow traffic. Each little clump of cars is a "characteristic," coasting along at its own local speed. The fast clump behind is gaining on the slow clump ahead the whole time — and when it catches up, the smooth density profile can't describe what happens next with a single value at a single point: cars pile up, and a sudden, sharp boundary appears between clear road and jam. That boundary is the shock, appearing out of nowhere even though every car obeyed a perfectly smooth rule the entire time.

Notice what's essential here and what isn't. It doesn't matter that the "stuff" is cars — swap in molecules of air, and "denser traffic moves slower" becomes "more compressed air pushes back harder," which is exactly the mechanism behind a shock wave in a gas. The specific substance changes; the mathematics of characteristics catching each other up does not.

Worked example: pinpointing where a shock is born

Take the decreasing initial profile u_0(x_0) = 2 - 0.5\,x_0 used in the figure below: taller (faster) values sit to the left, shorter (slower) values sit to the right — exactly the setup for a pile-up. The characteristic launched from x_0 is the line

Take two of them, starting at x_0 = a and x_0 = b with a \ne b, and ask when they meet:

Since a \ne b, the only way to satisfy this is 1 - 0.5t = 0, i.e. t = 2 — and that answer didn't depend on which pair a, b we picked! For this particular (linear) profile, every characteristic passes through the very same point at t = 2, namely x = 2t = 4. The whole fan of starting values collapses into a single instant and a single place: the shock is born at exactly (x, t) = (4, 2), precisely where the geometry below marks it.

The lines that crash together

Step through the picture: feet on the x-axis launch characteristics whose slopes are set by the initial height there. Because the left points are taller (faster), the lines lean in and meet — and the first crossing is where the shock is born, matching the algebra above exactly.

The physical reality

This is not a mathematical curiosity confined to a blackboard. Genuine shocks — sudden jumps that emerge from smoothly varying starting conditions — show up throughout physics and engineering:

• Sonic booms. A supersonic aircraft compresses the air ahead of it faster than sound can carry the disturbance away, and the compression waves pile up into the sharp crack you hear on the ground.
• Traffic jams. Exactly the mechanism above: a smoothly varying stream of cars develops a sudden density jump — the visible "wall" at the back of a jam — with no crash required to start it.
• Gas dynamics. Compressor blades, engine cylinders, and re-entering spacecraft all generate shock waves where a smooth pressure disturbance steepens into a near-discontinuous front.
• Water waves. A gently sloping ocean swell can steepen as it runs into shallow water, its front edge outrunning its back edge until the wave breaks — the same faster-catches-slower story, playing out in water instead of air or asphalt.

In every case, the ingredients are identical: a smooth starting state, a speed that depends on the local value of the very quantity being transported, and enough time for the fast parts to run down the slow parts ahead of them.