Pressure in shock compression

For people familiar with static compression techniques, equation of state is often expressed as f (P, V, T) = 0. This equation helps understand how these properties interact under different conditions, such as the Burch-Murnaghan equation of state (2nd or 3rd, commonly used for solids) . However, in the shock compression community, instead of volume and pressure, they use shock wave velocity (Us) and particle velocity (Up) to characterize the state of materials. This can seem different, but it's just another way to describe how materials behave under high-pressure conditions caused by shock waves.

Why (Us, Up) can represent a state of material?

When a material is subjected to a shock (a wave traveling faster than sound speed in that material), the collection of possible states of the material (P, V or ρ, T) lies on the Hugoniot line. One shock in the lab corresponds to one point on the line. In these experiments, two velocities are commonly measured, Us and Up. Pressure can be expressed as ρ₀×Us×Up. The determination volume and temperature in shock compression requires additional assumptions (Grüneisen's assumption + heat capacity of material).

Why P = ρ₀×Us×Up ?

Assuming conversation of momentum, we have F t = m(v-v₀), where m is the mass of material that the shock wave has traveled through in time t. m can be expressed as ρ₀× V.

Dividing both sides by the cross-sectional area S, we have P t = ρ₀× L × (v-v₀), where L represents the length that the shock wave traveled in time t.

In typical senarios where v₀ = 0, and note that L/t = Us and v=Up (the velocity this part of material has been accelerated to), we can simply to P=ρ₀×Us×Up.