Relativistic Hydrodynamics 2026, 08: Linear Hydrodynamic Waves

For a stationary, uniform perfect fluid in a flat spacetime under special relativity and energy-momentum conservation, then

We adapt first order perturbation notation and take the unperturbed fluid to be stationary and uniform.

Discontinuous waves can be categorized into contact waves (separating two parts of a fluid without flow through the surface) and in shock/reaction fronts (discontinuity surfaces, crossed by a flow). When a simple wave causes a decrease in pressure and rest-mass density in propagation regions, they are considered a "rarefaction" wave. They are isentropic solutions with a head (portion with largest velocity) and a tail (slowest portion). When choosing a comoving frame, the head is usually chosen as a reference. Rarefaction waves can be written in a self-similar form. A 1D self-similar solution depends on a similarity variable equal to x/t. At any point b behind the rarefaction head,

Shock waves can form compressive motions. They can be considered a planar discontinuity surface, dividing space into upstream and downstream of itself.

Cases where shock wave occur under absence of mass flux ([[J]] = 0) is a "contact discontinuity", from which follows [[p]] = [[h]] = 0, [[ρ]] ≠ 0, [[v]] = 0. They are trivial Riemann invariant, given by the pressure & velocity.

All nonlinear waves so far are solutions of the "Riemann problem", which determines the flow pattern in the presence of constant, discontinuous initial data. It can be described as an initial value problem with arguments x, t where for t = 0, the values for x above and below (arbitrarily) 0 take on different constants, commonly referred to as "left" and "right" states. Depending on the hyperbolic equations leads to different solutions. State switching results in a diagram of lines fanning out from the origin. This diagram is the "Riemann Fan". The generic 3-waves solution consists of three different wave patters. Either two shock waves, moving toward the left initial state, and the other toward the right initial state, one shock wave and one rarefaction wave, where the shock moves toward the right initial state, and the rarefaction toward the initial left, or vice versa, and two rarefaction waves in analogue to the first state.

General solutions to the Riemann problem can't be given in closed analytic form, but the numerical solutions can be considered exact. Based on initial conditions, the wave pattern is determined by a Newtonian approach. After identifying the relativistic frame-independent expression for the relative velocity between the two unperturbed states and comparing it with the three value limits transitioning between wave pattern, directly computed from the initial conditions. The continuity of pressure and velocity across discontinuity separating the outermost regions give a nonlinear root with a functional form different for each possible wave pattern. Once the pressures are known from the velocities, the solution of the Riemann problem can be completed by determining the other quantities in the end states.

Velocity jumps across a shock wave are no algebraically compatible. The ratio of the velocities ahead and behind the shock front needs to be found as roots of a nonlinear equation. For a propagation in x-direction, only the ratio between the y- and z-direction velocities is constant through the shock.