a rarefaction wave
A shock is a sudden squeeze — fast catching slow. A rarefaction is the opposite, a gentle spreading-out — slow ahead, fast behind, so the gap between them opens smoothly instead of closing into a jump. Where a shock concentrates, a rarefaction fans out. It is the smooth, continuous wave that fills in an expanding region.
Think of starting data with a jump that is 'expansive': u_L on the left smaller (in the sense that f'(u_L) < f'(u_R)) than u_R on the right, so the characteristic from the left moves slower than the one from the right and they fly apart, leaving an empty wedge in the (x, t) plane with no characteristics. A shock here would violate the entropy condition. Instead the solution fills the wedge with a continuous fan of values: for Burgers' equation, u(x, t) = x/t for u_L t < x < u_R t, a self-similar rarefaction depending only on the ratio xi = x/t. Each ray x/t = constant carries a single value, smoothly interpolating from u_L to u_R. In general the fan is built from the rarefaction curve, along which a Riemann invariant stays constant.
Why it matters: rarefactions and shocks are the two elementary building blocks from which the solution of every Riemann problem is assembled. A rarefaction is exactly what the entropy condition demands wherever a 'rarefaction shock' would otherwise be allowed by Rankine-Hugoniot — it is the physically correct way to resolve an expansion. Unlike a shock, a rarefaction is smooth for t > 0 (it even smooths a corner in the data), and it is reversible-looking, carrying no dissipation.
Burgers' equation with u_L = 0 on the left, u_R = 1 on the right (an expansive jump). The solution is the rarefaction u(x, t) = 0 for x < 0, u(x, t) = x/t for 0 < x < t, and u(x, t) = 1 for x > t. The initial step has smoothed into a ramp whose width grows linearly in time.
An expansive jump resolves into a self-similar fan u = x/t, smooth for every t > 0.
Whether a given jump becomes a shock or a rarefaction is decided by the entropy condition (equivalently by the sign of the speed change), not by Rankine-Hugoniot, which the rarefaction's endpoints would also satisfy. Compressive data shocks; expansive data rarefies.