# 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:

$\ P_1 - P_0 = U^2\left(\rho_0 - \frac{\rho_0^2}{\rho_1}\right)$

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## References

