text: First let us show that the relation  is not only valid near the weak discontinuity, but throughout the region. If this inequality is not maintained, then there must be a point where  between the weak discontinuity and the detonation front. The second governing equation implies that at this point  must be infinite or, . Let us obtain  by taking the second derivative of the governing equation. In the resulting equation, impose the condition  to obtain . This implies that  reaches a maximum at this point which in turn implies that  cannot exist for  greater than the maximum point considered since otherwise  would be multi-valued. The maximum point at most can be corresponded to the outer boundary (detonation front). This means that  can vanish only on the boundary and it is already shown that  is positive near the weak discontinuity,  is positive everywhere in the region except the boundaries where it can vanish. 
  Note that near the detonation front, we must satisfy the condition . The value evaluated at  for the function , i.e.,  is nothing but the velocity of the detonation front with respect to the gas velocity behind it. For a detonation front, the condition  must always be met, with the equality sign representing Chapman–Jouguet detonations and the inequalities representing over-driven detonations. The analysis describing the point  must correspond to the detonation front.