Linear Shock-Velocity-Particle-Velocity Relationship

An equation of state, based on a bulk modulus variation with pressure of the form $$B^S{\mathrm{=B}}_0^S{\mathrm{+B}}_0^{\mathrm{S\prime }}{\mathrm{P+B}}_0^{\mathrm{S″}}{P}^2\text{/2,}$$ where B₀^S, B₀^S′, and B₀^S″ are constants, is developed in this analysis. The resultant equation of state is combined with the Rankine‐Hugoniot conservation relations to obtain a Maclaurin series expansion for the shock velocity vs the particle velocity $$u_s{\mathrm{=c+su}}_p{\mathrm{+s\prime u}}_p^2\mathrm{+\cdots .}$$ The coefficients c, s, and s′ are given in terms of the unshocked density and quantities available from ultrasonic elastic constant measurements at high pressures. Using new experimental data for sodium, it is shown that s′ is nearly zero. For ionic crystals such as KBr a theoretical expression is given for B₀^S″ (in terms of B₀^S′). In the case of KBr, the value of s′ is also very close to zero. The smallness of s′ depends on the cancellation of a number of terms brought about by the fact that B₀^S″ is negative. CsI and xenon are also discussed.