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\end{equation}  Inserting equation~\ref{eqn:rho} to \ref{eqn:deriv} into eqn.~\ref{eqn:4}, we arrive at an ordinary differential equation, that describes the functional form of the shape of the shock front:  \begin{equation}  P(z) = \frac{3}{4}\rho_0\v_0^2 \frac{3}{4}\rho_0v_0^2  = \frac{3}{4} \frac{\dot{M}}{4\pi\v_{\inf}(z^2+\omega^2)} \frac{\dot{M}}{4\piv_{\inf}(z^2+\omega^2)}  v_{\inf}^2 \sin^2(\psi) \end{equation}  This equation can be simplified to  \begin{equation}\label{eqn:ode}