Approximate solution for spherical implosion of coalesced weak shocks in a plasma

An approximate analytical solution is obtained for the problem of spherical implosion of coalesced weak shocks in an ideal gas. Since the motion is assumed to be self-similar, our expressions are approximations to the exact solution which is valid close to the center of the sphere only. This has been achieved by replacing, in different time regimes, a nonlinear function of the reduced sound velocity and the reduced gas velocity appearing in the self-similar differential equation for these variables, by suitable functions of the reduced gas velocity alone. By considering a fully ionized D-T plasma to be an ideal gas of specific-heat ratio $\gamma =\frac{5}{3}$, it is shown explicitly that coalesced weak spherical shocks are much more efficient than a single strong spherical shock, in generating fusion energy.