Analysis of a cylindrical imploding shock wave

The self-similar solution of the gasdynamic equations of a strong cylindrical shock wave moving through an ideal gas, with γ = c_p/c_v, is considered. These equations are greatly simplified following the transformation of the reduced velocity \[ U_1(\xi)\rightarrow U_1 = {\textstyle\frac{1}{2}}(\gamma + 1)(U+\xi). \] The requirement of a single maximum pressure, d_ζP = 0, leads to an analytical determination of the self-similarity exponent α(γ). For gases with γ < 2 + 3^½ the slight maximum pressure occurs behind the shock front, nearing it as γ increases. For γ < 2 + 3^½, this maximum ensues right at the shock front and the pressure distribution then decreases monotonically. The postulate of analyticity by Gelfand and Butler is shown to concur with the requirement d_ζP = 0. The saturated density of the gas left in the wake of the shock is computed and − U is shown to be the reduced velocity of sound at P = P_m.