Consider a spherically outgoing Chapman–Jouguet detonation wave propagating with a constant velocity . By definition, immediately behind the detonation wave, the gas velocity is equal to the local sound speed  with respect to the wave. Let  be the radial velocity of the gas behind the wave, in a fixed frame. The detonation is ignited at  at . For , the gas velocity must be zero at the center  and should take the value  at the detonation location . The fluid motion is governed by the inviscid Euler equations
where  is the density,  is the pressure and  is the entropy. The last equation implies that the flow is isentropic and hence we can write .
Since there are no length or time scales involved in the problem, one may look for a self-similar solution of the form , where . The first two equations then become
where prime denotes differentiation with respect to . We can eliminate  between the two equations to obtain an equation that contains only  and . Because of the isentropic condition, we can express , that is to say, we can replace  with . This leads to
For polytropic gases with constant specific heats, we have . The above set of equations cannot be solved analytically, but has to be integrated numerically. The solution has to be found for the range  subjected to the condition  at  
The function  is found to monotonically decrease from its value  to zero at a finite value of , where a weak discontinuity (that is a function is continuous, but its derivatives may not) exists. The region between the detonation front and the trailing weak discontinuity is the rarefaction (or expansion) flow. Interior to the weak discontinuity  everywhere.