Adiabatic shock wave equation

For a transmitted shock wave advancing into any gas at an initial pressure pe of 1 atm, the RH (Rankine-Hugoniot) equation defines a functional relationship between pressure p and particle velocity w behind the wave S3, involving initial pressure, initial specific volume v, and equations of state of the target medium. Similarly, the conditions behind the reflected wave S2 and close to the product-target interface are expressible by means either of the shock wave equations or the Rie-mann adiabatic wave equations in terms of any one such variable and the conditions... 

Once the shock wave pressure and the shock wave velocity in a material are determined experimentally, the adiabatic shock (Hugoniot) equation of this material may be defined in the form p=J U ). Also, if the mass velocity and the shock wave pressure (or shock wave velocity) are being determined experimentally in a tested material, the adiabatic shock equation may be defined in the formf/ ,=y(ir )or p=AWm). 

It has been seen in deriving equations 4.33 to 4.38 that for a small disturbance the velocity of propagation of the pressure wave is equal to the velocity of sound. If the changes are much larger and the process is not isentropic, the wave developed is known as a shock wave, and the velocity may be much greater than the velocity of sound. Material and momentum balances must be maintained and the appropriate equation of state for the fluid must be followed. Furthermore, any change which takes place must be associated with an increase, never a decrease, in entropy. For an ideal gas in a uniform pipe under adiabatic conditions a material balance gives ... 

In solving an actual problem, Paterson advises to base the choice of shock or adiabatic wave equation for the products on an assumption regarding the nature of S2. If the conclusions contradict the assumption, the alternative equation is to be used. Although the general criterion for shock reflection is thus of a posteriori nature, it can be expressed in the form ... 

A sample calculation was given in Ref 29b for shock waves in air employing the general equation of state (See eq 23) for the expln products and the ideal gas adiabatic eq of state for air. Employing the method of calculation based on the hydrodynamic theory, good agreement was obtd for expls investigated (PETN Tetryl) between calculated and experimentally detd explosive properties... 

The equations describing a chemical explosion are the same as those for a mechanical explosion (see Sec. 5.3) that is, to compute the maximum energy released in an explosion, we assume the process within the region bounded by the shock wave is reversible and adiabatic so that... 

Equation F.12 is not specific for shock waves but applies to any steady, adiabatic flow of a perfect gas. The reader may check a few V7c values in App. A.5 to see that they have indeed been made up from Eq. F.12. 

Tlie calculation of the detonation pressure is based on the fact that the pressure and the rate of the material flow behind the passing and the reflected waves are equal on both sides of the contact surface. The impulse conservation law and the known metal plate adiabatic shock equation are applied. 

The aquarium test is essentially a modification of the flying plate test in which the metal plate is replaced by a layer of water or some other optically transparent material, such as Plexiglas. Instead of the determination of the plate free-surface velocity, the traveling of the shock wave through an inert and optically transparent material is viewed as a function of time. When having the shock wave velocity vs. distance dependence, and knowing the adiabatic shock equation of the inert material used, the mass velocity behind the shock wave front and the detonation pressure of a tested explosive may be calculated. 

The mass velocity is calculated from the known shock wave velocity and known adiabatic shock equation for PMMA. The latter can be written in the form (Ashaev et al., 1987) ... 

If the CJ point is not reached, i.e., there is no marked change in the slope of the curve, then the calculation of the detonation parameters is based on the extrapolation of the shock wave velocity to the zero distance (initial shock wave velocity) and consequent calculation of the corresponding mass velocity and pressure, applying the adiabatic shock equation and Goranson equation (the calculation procedure is similar to the one used in the flying plate test). 

Besides, the flow conditions in the funnelled cylinder are also defined by the Bernoulli equation. The equation predicts that in the stationary flow conditions, the sum of the enthalpy (h=e+pV) and kinetic ener [(D-Wf/2] is constant along the flow path. For the isentropic expansion, on the other hand, it is valid that dh= Vdp. Together with the equation of conservation of energy in the detonation wave (i.e., the adiabatic shock equation), this gives the equation for the energy conservation for the detonation products flow conditions in the metal cylinder ... 

It is evident from the above that, in order to establish the adial tic shock equation for a given medium, at least two shock wave parameters have to be determined experimentally. Modern experimental methods enable direct determination of the shock wave pressure and shock wave firont velocity with good accuracy. This proves to be sufficient for defining the adiabatic shock equation for a given medium. Once the adiabatic shock equation of a material is known, one may predict the behaviour of the material under cfynamic pressure action. 

The Equation of State of Detonation Products Behind Overdriven Detonation Waves in Composition B (Ref 15, pp 47-52), stated that experiments conducted in England have shown that in many high explosives the shock compressions and adiabatic expansions of... 

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