Properties of the Hugoniot function

Differentiation of the Hugoniot function H (p,v) = E(p,v)-E(p₀,v₀)+1/2(p + p₀)(v₀ - v) and use of the first and second laws of thermodynamics leads to the relation dH = T dS+dA, where dA is the element of area in the (p, v) plane swept out (in a counter-clockwise direction) by the line segment (p0, v0) → (p, v) as the point (p, v) is moved from some point (p1, v1) to a neighbouring point (p1+dp1, v1+dv1). This relation, together with rather general assumptions regarding the shape of the isentropic curves dS = 0 for the material behind the shock, makes possible the geometrical derivation of a number of properties of the function H and of the Hugoniot curves dH = 0.