A wave nothing in it travels with

Take any quantity that is conserved and whose flux depends on nothing but its own local density:

$$\frac{\partial c}{\partial t} + \frac{\partial q(c)}{\partial x} = 0,\qquad q = q(c).$$

That is the whole model. There is no momentum equation in it, no pressure, no viscosity and no inertia. The only physics anywhere is the choice of the function $q(c)$, and everything below follows from its shape.

Rewriting it as $\partial c/\partial t + q'(c)\,\partial c/\partial x = 0$ says that $c$ is carried at speed $q'(c)$ — the slope of the flux curve — while the material itself moves at $q/c$, which is the chord from the origin. Those are different numbers for every flux law that is not a straight line, and the difference is the subject.

The river #

For a wide channel, Manning’s formula makes the discharge per unit width go as the five-thirds power of the depth: $q = kc^{5/3}$. Then

$$\frac{q'(c)}{q(c)/c} = \frac{5}{3}\quad\text{exactly, at every depth.}$$

Not approximately, not for shallow water, not for a particular roughness. The ratio is the exponent, and the exponent is a property of the friction law rather than of the river. Measured across a sweep of depths it holds to $2\times10^{-16}$.

So a flood crest travels at five-thirds the speed of the water in it. After eight time units the crest is $3.58$ ahead of a marked parcel of the water that raised it.

That is why flood warning is possible. The rise arrives before the water, and the lead time is the difference between two straight lines.

Traffic, where the sign changes #

Greenshields’ fundamental diagram — $q = v_f c(1 - c/c_j)$, a parabola through the origin and the jam density — gives a wave speed $v_f(1 - 2c/c_j)$ and a vehicle speed $v_f(1 - c/c_j)$.

The second is positive at every density: nobody is driving backwards. The first changes sign at $c = c_j/2$, exactly, and is $-0.8v_f$ near the jam.

A queue that appears from nowhere is a wave running backwards. Every driver in it is going forwards, the disturbance is going backwards, and there is no contradiction because a wave is not a thing being carried.

The steepening, and where it starts #

Because different values of $c$ travel at different speeds, a smooth profile does not stay smooth. Every characteristic is a straight line $x = x_0 + q'(c_0(x_0))\,t$, and the first crossing is at

$$t_b = -\frac{1}{\min_x\ \mathrm d[q'(c_0)]/\mathrm dx}.$$

For a Gaussian hump on a background depth, that is $t = 3.044$ — and the point at which it happens is $x = 0.90$, which is on the front face of the hump rather than at its crest. Nothing in the initial condition is discontinuous, and the solution becomes vertical anyway.

That is the same mechanism a finite-amplitude sound wave uses, and it is worth having both, because the two essays own different halves of why it happens.

What the front travels at #

After the crossing the conservation law is satisfied by a discontinuity. Its speed is not either of the two characteristic speeds; it is the chord between the states on the two sides,

$$s = \frac{[q]}{[c]},$$

which is the Rankine–Hugoniot condition arrived at from conservation with no momentum equation anywhere.

Solving the equation numerically with a scheme that knows nothing about shocks — a local Lax–Friedrichs flux on four thousand cells, first order and deliberately so — gives a front whose measured speed is the chord slope of the states either side of it to two parts in a thousand.

Why the chord, and why that makes the front stable #

The two tangents in that figure are worth a paragraph, because they say why the discontinuity persists.

The characteristic speed behind the front is larger than the front’s speed, and the one ahead is smaller. So characteristics run into the discontinuity from both sides and are absorbed by it. That is the entropy condition, in the form Lax stated it, and it is the reason a compression steepens into a front and stays as one.

Run the same argument backwards on an expansion — a step down in depth — and the characteristics diverge from the discontinuity instead of running into it. No such front exists; the step spreads into a fan. That asymmetry between compression and expansion is the deepest thing in the compressible field and it appears here with no gas in it at all.

The condition that is not in the equation #

The front is selected above by an argument about which way the characteristics run, and it is worth saying plainly that this is an extra condition rather than a consequence of the conservation law. The law by itself does not have one answer.

Take a downward step — the reverse of the front. It satisfies the conservation statement exactly, its speed is given by the same chord rule, and nothing in $\partial c/\partial t + \partial q/\partial x = 0$ forbids it. It is a perfectly good weak solution, and there are infinitely many others: the equation admits solutions with discontinuities placed almost anywhere, and it has no way to prefer between them. What is missing is a rule saying which of them a physical system takes.

The gas-dynamics answer is thermodynamic — the expansion shock is refused because it lowers entropy — and here it cannot be, because there is no thermodynamics anywhere in a queue of cars. The condition is still called an entropy condition, and the name is borrowed from a subject that has no bearing on this one.

What actually selects the answer is the term the model deleted. Add a small diffusion, $\varepsilon c_{xx}$, solve, and let $\varepsilon$ go to zero: the limit exists, it is unique, and it is the solution with the fronts the right way round. The downward step is not in the limit, because with any diffusion at all it spreads immediately and never re-forms.

So this collection’s standing observation applies once more: setting a small parameter to zero and letting it go to zero give different answers, and the extra information survives the limit as a selection rule. It is the same shape as the vorticity inside a closed streamline, one field away.

And there is a practical edge on it. A numerical scheme’s artificial dissipation is doing the selecting, so a scheme that is conservative and convergent can converge to the wrong weak solution — producing a clean, stable, mesh-converged expansion shock that satisfies every conservation check and does not happen. Schemes are checked against exactly that case.

What is not in the flow #

The constitutive law. Kinematics supplies the conservation statement and nothing else; $q(c)$ is a piece of physics imported whole — Manning’s formula for the river, a measured fundamental diagram for the traffic, a settling velocity for a suspension, a permeability for a bed.

Change it and every speed on the page changes while the equation does not. That is unusually visible here because the two examples share the equation exactly and share no number: one has a ratio of $5/3$ at every density and the other has a wave speed that changes sign.

It is also why the model is portable. The same three lines describe flood waves, traffic, sediment transport, glacier flow, chromatographic fronts and the movement of a shock in a fluidised bed, and the only thing that has to be supplied for each is a curve.

Where the wave speed is not the flux slope #

The result is worth stating with its hypothesis, because the hypothesis is what makes it a kinematic wave rather than a dynamic one.

$q$ must depend on $c$ and on nothing else — not on $\partial c/\partial x$, not on the history, not on $t$. That is a strong assumption. A real river’s discharge depends on the local slope of the water surface as well as its depth, which adds a $\partial c/\partial x$ term and turns the equation into a diffusion-modified one; the wave then spreads as it travels and the sharp front is a limit rather than an outcome.

The dimensionless number governing which case applies is the ratio of the kinematic wave speed to the dynamic wave speed $\sqrt{gh}$ — that is, the Froude number — and kinematic wave theory is the low-Froude limit. Above it the flow has its own gravity waves and both mechanisms are present.

The relation to the site’s other shocks #

Three appearances of the same structure, worth putting side by side.

The shock in a river is a hydraulic jump, which has a momentum equation in it: the conjugate depths come from conserving momentum flux, not from a flux law. That is a dynamic shock and its speed involves $\sqrt{gh}$.

A kinematic shock is this essay’s front. Its speed involves only the flux curve, and it exists in systems with no momentum at all — a queue of cars has no momentum flux in any useful sense.

A gas shock is dynamic, and its jump condition is the same chord rule applied to three conserved quantities at once rather than to one.

All three obey $s = [q]/[c]$ for whatever is conserved. What differs is how many things are conserved and where the flux law comes from.

Where else the same three lines apply #

The portability is worth spending a section on, because it is the strongest argument for taking the model seriously and it is also where its limits show.

Sediment. The flux of sand past a station depends on the local bed shear, which depends on the local depth, which depends on the local bed elevation. That closes into a conservation law for the bed with a flux depending on the bed itself, and the resulting waves are dunes and antidunes. They migrate at a speed unrelated to either the water or the grains, and whether they go downstream or upstream is the sign of a derivative — the same sign question the traffic case answers at $c_j/2$.

Chromatography. A solute moving through a packed column partitions between the mobile and stationary phases, so its effective flux depends on its own concentration through an adsorption isotherm. The front of a band steepens or spreads according to whether the isotherm is convex or concave, and the whole practical art of the technique is choosing conditions in which it does the useful one.

Glaciers. Ice flux depends on thickness through a power law with an exponent near four, so a kinematic wave in a glacier travels at four times the ice velocity. That is why a change in accumulation at the head of a glacier reaches the snout long before the ice does.

Two-phase flow in a pipe. The drift-flux model is exactly this equation, and the void-fraction waves it predicts travel at speeds that have nothing to do with either phase’s velocity.

In every one of these the equation is the same and the curve is different, and getting the curve is the whole experimental programme.

Reading the flux curve #

Since everything is in the curve, it is worth saying what to look for in one.

The slope at the origin is the wave speed of a vanishingly small disturbance on an empty system — a ripple on a dry bed, a car on an empty road.

Where the slope is zero is where a disturbance does not move at all. For traffic that is the density of maximum flow, which is why a road at capacity has stationary disturbances sitting in it and why capacity is where a road becomes fragile.

Where the curve is convex the compressions steepen and the expansions spread; where it is concave the reverse. Manning’s law is convex everywhere, so a rising flood steepens and a falling one spreads out — which is why hydrographs are asymmetric, with a sharp rise and a long recession, and the asymmetry is purely a property of the exponent.

And the chord to the origin is the material speed, always. That is the one line on the diagram that carries no information about disturbances at all.

Why the momentum equation is missing #

It is worth being explicit about what has been left out, because the model’s success at doing without it is surprising.

A momentum equation would say how fast the fluid accelerates when there is an unbalanced force on it. Kinematic wave theory replaces that with an assumption — that the flow is always in local equilibrium, so the discharge is whatever the local depth alone would produce in a steady uniform flow. That is a statement that the flow adjusts to its local state faster than the wave moves through it, and it is the same kind of assumption as a quasi-steady aerodynamic model makes about an oscillating wing.

When it holds, the momentum equation has been used once, offline, to produce the curve, and then never again. When it does not hold — a flood wave in a steep channel, a traffic wave with strong driver anticipation — the dynamics come back and the wave speed acquires terms the flux curve does not contain.

The check is the same in every application: compare the time the flow needs to adjust locally with the time the wave takes to pass. That is a ratio of two timescales, which is the shape every regime question on this site takes.

What a crest is not #

One consequence worth stating flatly, because it is the reading the essay’s title is about.

The crest of a flood wave is not a body of water. It is not a mass that travels. It is a place where the depth is a maximum, and the water in it at one instant is different water from the water in it an instant later — the wave has moved forward through fluid that is moving forward more slowly.

The same is true of a traffic jam: the vehicles in a queue at one moment are not the vehicles in it a minute later, and the queue’s back end moves upstream through cars that keep arriving and leaving. And it is true of a sand ripple, which migrates downstream at a fraction of the grain velocity while the individual grains are being lifted over its crest and dropped on its lee.

Nothing in the wave travels with the wave. That is the whole of the title, and it is the property that makes these disturbances confusing to watch and easy to compute.

The model limit #

The whole framework holds for a single conserved quantity with a flux depending on nothing else. Three places where that fails:

A non-convex flux. If the second derivative of $q$ changes sign — which happens for sediment, and for traffic models with more structure than Greenshields’ — the solution can contain composite waves that are part shock and part fan, and the chord rule alone does not determine the answer.

Several quantities at once. Two conserved densities give two characteristic families, two wave speeds, and interactions between them, which is exactly what a shock tube does with its three.

And the front itself. The discontinuity here is a limit of the model, not a physical surface. In a real river it is a bore a few metres long; in traffic it is a deceleration lasting seconds; in a gas it is a few hundred nanometres of viscous structure. The kinematic model says where the front is and how fast it goes, and it says nothing at all about what is inside it — which is the same division of labour the jump conditions and the shock structure have in the compressible field.

There is a fourth limit which is about the arithmetic rather than the physics, and it is worth stating because the figures depend on it. The numerical solution here is first order, so the front it produces is smeared over several cells, and the states used to compute the chord slope are read forty cells either side of it. Reading them too close in gives the smeared values and a wrong chord; reading them too far out gives states that have themselves changed, since the hump is still spreading. The measurement is made between two snapshots a fifth of a time unit apart for the same reason: the front decays as it runs, so its speed is a function of time, and averaging over a long interval compares a mean with an instantaneous prediction and reports the difference as an error.