A supersonic stream of a perfect gas turns through a fan of infinitesimal Mach waves

PHYSICS · Subjective · Fermi University Physics · Mathematical Methods for Physics · Fluids & Continuum Physics · Calculus of change and accumulation

Signal Speeds in Fluids, Ch Oblique shocks, expansion fans and blast waves

Problem

A supersonic stream of a perfect gas turns through a fan of infinitesimal Mach waves, each turning it by dθ=M2−1 duu\mathrm{d}\theta = \sqrt{M^2-1}\,\dfrac{\mathrm{d}u}{u}, where uu is the speed and MM the local Mach number. Using the isentropic relation between speed and Mach number for a perfect gas with ratio of specific heats γ\gamma, and taking ν(1)=0\nu(1)=0, integrate this relation to the Prandtl-Meyer function ν(M)=γ+1γ−1 arctan⁡ ⁣[γ−1γ+1(M2−1)]−arctan⁡ ⁣(M2−1).\nu(M) = \sqrt{\frac{\gamma+1}{\gamma-1}}\,\arctan\!\left[\sqrt{\frac{\gamma-1}{\gamma+1}(M^2-1)}\right] - \arctan\!\left(\sqrt{M^2-1}\right). Show the key substitution and the final integral explicitly; do not simply quote the result.

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