I'm studying the Riemann problem in hydrodynamics. I've learned that a rarefaction wave and a shock wave will be created.
From what I know, a shock wave is a wave which travels at higher speeds than the speed of sound in that medium. However, I can't find a way to fit my definition to the problem, as there is no need for the wave to travel at such speeds in the Riemann problem. Therefore, I would like to ask what is exactly a shock wave in this field of physics.
Formally, the Riemann problem does not say anything about shock waves. It is just an initial value problem of the form: $$ q\left( x, t=0 \right) = \begin{cases} q_{L} & \text{for x $\leq$ 0} \\ q_{R} & \text{for x $>$ 0} \tag{0} \end{cases} $$ where $x$ is a spatial variable and $t$ a temporal variable. The problem states that at time zero, all values of $q$ are constant for negative $x$ and all values of $q$ are constant for positive $x$, but they transition between these two constant states in some region or a discontinuity, depending on the problem you wish to address.
The problem is ideally suited for 1D gas dynamics, e.g., there is a removable wall at $x$ = 0 and two gases on either side with different densities and temperatures.
In the case of full hydrodynamics to which you refer, the dummy variable $q$ represents the following multiple variables: density, pressure, and bulk flow velocity (or speed for 1D problem). The Riemann problem is often used as a useful tool for initial value problems involving shock waves, e.g., shock tube problems are ideally suited for this.
However, I can't find a way to fit my definition to the problem, as there is no need for the wave to travel at such speeds in the Riemann problem. Therefore, I would like to ask what is exactly a shock wave in this field of physics.
The resulting discontinuities may or may not be shock waves. If the initial conditions have a large enough gradient between the two regions for the parameters, a shock wave would form. The gradient is small, then a simple acoustic wave will form. If proper dissipation (e.g., viscosity) is included, the initial discontinuity sound wave will smooth out (i.e., decrease in amplitude/strength) into a gradual change in the absence of external work.
The whole shock thing arises from large pressure/flow/density gradients that cause large enough forces to maintain a shock. The initiation of the shock requires some irreversible energy transformation (in thermodnyamics, this is equivalent to entropy production).
