Strong Convergent Shock Waves Near the Center of Convergence:A Power Series Solution

A Lagrangian formulation of the self-similar convergent shock problem is presented for a quiescent perfect gas of zero pressure. The initial density is either constant or decreasing towards the center according to a power law. One-dimensional plane, cylindrical, or spherical geometry is assumed. The paper is based on a power series representation of the nondimensional flow velocity f(τ ). Density, pressure, and particle trajectories are expressed analytically by f. Similarity parameters α, introduced as the near-center limit of a function describing the motion of the shock front, are determined by requiring the power series of f to converge. Defining characteristic polynomials Pₙ (α ) by the derivatives f⁽ⁿ⁾ (1) generated from an ordinary differential equation, α is given by the limiting value of a converging sequence \ α ₙ | n = 1, ⋯ \ of real zeros of Pₙ (α ). Similarity parameters α ≈ α ₙ are given to 15 significant decimal digits.