Engineering Acoustics/The Rankine-Hugoniot Jump Equations

Conservation Equations and Derivation

For the purpose of performing engineering calculations, equations linking the pre- and post- shock states are required. One of the most fundamental expressions relating states is the $(P,\rho )$ Hugoniot which relates pressure and density as:

$\ P=f(\rho )$

This expression can be derived via simplification of the canonical conservation equations:

Conservation of mass:

$\rho _{1}\nu _{1}=\rho _{o}\nu _{o}\,$

Conservation of momentum:

$p_{1}+\rho _{1}\nu _{1}^{2}=p_{o}+\rho _{o}\nu _{o}^{2}$

Conservation of energy:

$e_{1}+{\frac {p_{1}}{\rho _{1}}}+{\frac {1}{2}}\nu _{1}^{2}=e_{o}+{\frac {p_{o}}{\rho _{o}}}+{\frac {1}{2}}\nu _{o}^{2}$

The parameters of a shock required to completely solve for the jump conditions are pressure, particle velocity,specific internal energy,density and shock speed. With 4 state variables and only 3 equations an additional equation is required to relate some of the states and make the problem tractable. This equation is referred to as an equation of state (EOS) - of which many exist for a variety of applications. The most common EOS is the ideal gas law and can be used to reduce the system of equations to the familiar Hugoniot expression for fluids with constant specific heats in steady flow:

${\frac {p_{1}}{p_{2}}}={\frac {(\gamma +1)-(\gamma -1){\frac {\rho _{2}}{\rho _{1}}}}{(\gamma +1){\frac {\rho _{2}}{\rho _{1}}}-(\gamma -1)}}$

For general, non-linear elastic materials there exists no equation of state that can be derived from first principles. However, a huge database of experimental data has revealed that virtually all materials display a linear relationship between particle velocity and shock speed (the voracity of the linear assumption in the method of characteristics example is now even more clear!):

$\ U=C_{o}+s\nu$

This equation is also known as the shock Hugoniot in the $U-\nu$ plane.

Combination of this linear relation with the momentum and mass equations yields the desired expression for the Hugoniot in the $P-\rho$ plane for virtually all solid materials:

$\ P=C_{o}^{2}{\frac {{\frac {1}{\rho _{0}}}-{\frac {1}{\rho }}}{\left({\frac {1}{\rho _{0}}}-{\frac {s}{\rho }}\right)^{2}}}$

Paths and Jump Conditions

The Hugoniot describes the locus of all possible thermodynamic states a material can exist in behind a shock, projected onto a two dimensional state-state plane. It is therefore a set of equilibrium states and does not specifically represent the path through which a material undergoes transformation.

Consider again our discussion of strong and weak shocks. It was said that weak shocks are isentropic and that the isentrope represents the path through which the material is loaded from the initial to final states by an equivalent wave with converging characteristics (termed a compression wave). In the case of weak shocks, the Hugoniot will therefore fall directly on the isentrope and can be used directly as the equivalent path.

In the case of a strong shock we can no longer make that simplification directly, howevewer for engineering calculations it is deemed that the isentrope is close enough to the Hugoniot that the same assumption can be made.

If the Hugoniot is approximately the loading path between states for an "equivalent" compression wave, then the jump conditions for the shock loading path can be determined by drawing a straight line between the initial and final states. This line is called the Rayleigh line and has the following equation:

$\ P_{1}-P_{0}=U^{2}\left(\rho _{0}-{\frac {\rho _{0}^{2}}{\rho _{1}}}\right)$