I've read quite a few things about rarefaction waves in terms of giving weak and entropy solutions to certain PDE problems with fixed initial time data. I understand that the consideration of these solutions provide some sense of density that decreases with the passing of time, but I've not heard much else. My concerns are:
Why are these kind of solutions specially noticed as one of the very first examples given in several books?. Maybe it's some sort of "unwritten" but written tradition to expose these solutions.
Are they directly applicable in some specific physics topic or something? I'm pretty sure they are since every attempt I've done to try to research about the topic results in some content including the word "density".
Any thought, suggestion or reference could be very helpful to me ;)
Why are rarefactions emphasized?
Rarefaction solutions and shock solutions admit a complete characterization of the solution to the Riemann problem in one spatial dimension for a wide class of nonlinear hyperbolic conservation laws. Riemann solutions are of the utmost importance in applications as piecewise constant approximations to, e.g. flow fields around an aircraft, are use to discretize complex systems.
Are rarefactions important in real life?
Rarefactions play an important role in fluid dynamics. In fact, the Lax entropy condition is a generalization of the Second Law of Thermodynamics as it applies to compressible fluid flow. One can show that under mild assumptions of a compressible flow (isentropy) that thermodynamic entropy can 1. never decrease (2nd Law of TD) and 2. can only ever change at a shock. This allows for one to compute the change in entropy over a shock and determine that shocks can only occur in a single "direction." That is, one cannot have fluid discontinuously accelerate, or density discontinuously decrease. This is the equivalent to a shock in the middle of a rarefaction fan, which cannot occur via the Lax entropy condition (thermodynamic entropy is a Lax entropy for the Euler equations).
This matches up with experiments, where one can set up discontinuous initial conditions for a fluid via some apparatus, e.g., a shock tube, and observe rarefaction waves with no intermediate discontinuities when the initial conditions match that of a mathematical rarefaction, and true shock waves when the initial conditions match that of a shock given by the Rankine-Hugoniot conditions. The shocks we see in real life are known as compression shocks and they typically involve high-speed fluid colliding downstream with low-speed fluid, creating a disctoninuiyt in the mathematical idealization. If we didn't have the Lax entropy condition that guaranteed the uniqueness of rarefaction solutions, then we could have the opposite situation where low-speed fluid instantly accelerates to a higher speed, known as an expansion shock. Expansion shocks can occur in some numerical discretization schemes where the exntropy conditions is not explicitly enforced and stabilization techniques are required to ensure that physical shocks are correctly captured and that expansion shocks do not occur, resulting in rarefaction solutions according to the Lax entropy condition.