An aircraft travelling faster than sound creates a shockwave.
How dense is the air in this shockwave, and how thick is the shockwave?
Is this a range or a definite number? I realize the shockwave will change as the distance from the aircraft increases, but for close to the aircraft is sufficient.
How to go about calculating/researching these numbers?
How dense is the air in this shockwave...?
The density, pressure, temperature, and speed changes across a shock are given by the Rankine-Hugoniot conservation relations. If we use the subscripts $up$ and $dn$ for upstream(pre-shock) and downstream(shocked), respectively, for various parameters then one can show: $$ \frac{ \rho_{dn} }{ \rho_{up} } = \frac{ \left( \gamma + 1 \right) M_{up}^{2} }{ \left( \gamma + 1 \right) + \left( \gamma - 1 \right) \left( M_{up}^{2} - 1 \right) } = \frac{ U_{up} }{ U_{dn} } $$ where $\rho_{j}$ is the mass density in region $j$, $M_{j}$ is the Mach number in region $j$, $U_{j}$ is the bulk fluid flow speed along the shock normal unit vector in region $j$, and $\gamma$ is the ratio of specific heats.
...and how thick is the shockwave?
The thickness of a shock wave, if I am interpreting your question correctly, is determined by the dissipation mechanism (e.g., how you convert bulk flow kinetic energy into heat). In a neutral collisional fluid like Earth's atmosphere, the dissipation mechanism is binary particle collisions. Thus, the shock ramp thickness is roughly the mean free path, which is roughly a micron in Earth's atmosphere.
Is this a range or a definite number? I realize the shockwave will change as the distance from the aircraft increases, but for close to the aircraft is sufficient.
I am not sure what you are asking but if the aircraft is flying at a constant speed relative to the background fluid, the standoff distance (i.e., location of shock) will not change unless the incident background fluid changes. The standoff distance depends upon the shock geometry (e.g., see my answer at https://physics.stackexchange.com/a/271088/59023).
How to go about calculating/researching these numbers?
I would look into books on gas dynamics and shock waves (there are lots of them). For instance, Frank H. Shu has a book entitled The Physics of Astrophysics Volume II: Gas Dynamics that has some good discussions. If you are looking for more mathematically rigorous and fundamental derivations of parameters, then you would want to look for G.B. Whitham's book Linear and Nonlinear Waves.
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