Revisiting interaction of linear waves with an oblique shock: a spatial analysis and the inverse problem

The interaction of linear waves with a planar shock is analysed using a spatial framework. Unlike conventional temporal analyses which often overlook post-shock evanescent waves, our spatial framework intrinsically captures all wave modes. By prescribing the frequency and tangential wavenumbers, the shock-normal wavenumbers are solved simultaneously on both sides, yielding a complete wavevector correspondence across the shock. Based on this relationship, a linear system is established that links the amplitudes of all six incident waves (from either side) to five downstream responses: the four generated post-shock waves plus shock deformation. This system enables the solution of both the forward problem (predicting downstream responses from given incident waves) and the inverse problem. Specifically, by imposing no incident downstream acoustic wave, the inverse solution uniquely determines the required upstream waves to produce any desired downstream wave field. We further demonstrate that while the tangential base flow velocity alters the group velocity and frequency via the Doppler effect, it leaves the disturbance wavenumbers and amplitudes unchanged. Consequently, oblique shock interactions can be reduced to equivalent normal shock problems. As an application, we present a complete solution map for a Mach 2 normal shock flow. The complex amplitudes of evanescent waves reveal characteristic phase shifts relative to incident waves. Notably, we present a scenario where shock oscillations occur without any transmitted waves downstream, demonstrating that the shock can act as a perfect energy absorber. Finally, direct numerical simulations for three representative cases provide quantitative validation of these theoretical predictions.