Gradient relations at the front of shock and detonation waves

A natural assumption on the form of the calorific equations of state (internal energy) for one-dimensional motion was used to obtain the so-called gradient relations that give a one-to-one correspondence between the first partial spatial derivatives of the pressure, density, mass velocity (gradients of parameters) at shock and detonation fronts and the time derivative (acceleration) of the front. The assumption is based on the fact that, taking into account the thermal equation of state, the total internal energy, including both the thermodynamic part and potential chemical energy, can be represented as a function of pressure and density. This holds for both inert media and reaction products in the state of chemical equilibrium.