A Note on the Matched Asymptotic Structure of Weak Shock Reflection at Nearly Glancing Incidence

Abstract

We study the reflection of a weak planar shock from a rigid wall in the joint limit of weak shock strength and nearly glancing incidence. In the distinguished scaling (M=1+λα^2), where (M) is the incident-shock Mach number and (α) is the glancing angle, the inner reflection region is governed by the unsteady transonic small-disturbance (UTSD) equation. The corresponding canonical shock-reflection problem is controlled by the single parameter[a=\fracα{\sqrt{2(M^2-1)}}=\frac{1}{2\sqrtλ}+O(α^2),]so the limiting inner parameter (a_0=1/(2\sqrtλ)) is independent of (γ). Consequently, the detachment value (a_d=\sqrt2) maps to the physical scaling threshold (λ_d=1/8), with Guderley--Mach reflection for (λ>1/8). The physical trajectory angle is obtained from the canonical UTSD trajectory function (g(a)) by the Mach-number strength scale[χ_{\rm phys}\sqrt{2(M^2-1)},g(a)+O(M^2-1) 2\sqrtλ,α,g(a_0)+O(α^3).]We derive the self-similar UTSD reduction, the sonic parabola, the UTSD shock polar and its regular-reflection cubic, recovering (a_d=\sqrt2) directly. We also give the local linearisation and formal adjoint solvability condition defining the first correction (H(a;γ)), without claiming a computed correction curve. Finally, a time-marching solver for the full leading-order canonical UTSD system is benchmarked against the Hunter--Tesdall (a_0=0.5) triple point: once transverse compression (u>1) behind the Mach stem is retained, the computed (u=0.5) contour passes through ((ξ,η)=(1.007,0.514)), compared with the published ((1.008,0.514)).