Sirri Oguz – All About Pyrotechnics
LESSON 02 / DETONATION PHYSICS
Shock Waves and Conservation Laws
Fundamentals of Shock Waves
A shock wave is a propagating compression disturbance across which pressure, density, temperature, and material velocity change abruptly. Ordinary sound waves represent relatively small pressure disturbances moving near the local sound speed. A shock wave involves nonlinear compression and an irreversible entropy increase.
2.1 Shock Velocity and Particle Velocity
Shock velocity, Uₛ, is the speed at which the front advances through initially undisturbed material. Particle velocity, uₚ, is the material motion induced behind that front. The two quantities are distinct.
For a one-dimensional shock advancing into material initially at rest, continuity gives:
Here ρ₀ and ρ₁ are initial and shocked densities, respectively. Equation (2.1) expresses continuity of mass across the moving shock.
Physical Interpretation
Consider a shock wave propagating through an initially stationary aluminum plate. As the shock travels through the material, it compresses the aluminum and causes the material behind the shock front to move.
Two distinct velocities describe this process:
- Shock velocity (Uₛ) is the speed at which the shock front propagates through the initially stationary material.
- Particle velocity (uₚ) is the velocity acquired by the material immediately behind the shock front.
Although both velocities are directed along the direction of shock propagation in this example, they are not equal. The shock front advances faster than the compressed material moves behind it.
This distinction is fundamental to shock-wave physics and is essential for understanding the Rankine–Hugoniot conservation relationships.
Figure 2.1. Shock Velocity and Particle Velocity Across a Moving Shock. Schematic representation of a one-dimensional shock propagating through initially stationary material. The shock front advances at velocity (Uₛ), while material behind the front moves at particle velocity (uₚ). The initial and shocked states are denoted by subscripts 0 and 1, respectively. Velocities are shown in the laboratory reference frame.
2.2 Compression and Pressure
The shock increases density and generally raises pressure and temperature. For a compressive shock in a conventional medium:
Shock compression is an irreversible thermodynamic process. Unlike an ideal isentropic compression, entropy increases across a physical shock wave. This distinction becomes important when comparing the shock Hugoniot with an isentropic compression path in Lesson 03.
2.3 The Rankine–Hugoniot Relations
The Rankine–Hugoniot relations connect the thermodynamic and flow states immediately ahead of and behind a steady, one-dimensional shock wave through conservation of mass, momentum, and energy.
The following equations are expressed in a shock-fixed reference frame, in which the shock front remains stationary while material flows through it. For this introductory treatment, the shock is assumed to be nonreacting, with no significant external energy transfer across the discontinuity.
2.3.1 Conservation of Mass
The velocities w₀ and w₁ are measured relative to the shock; m is mass flux per unit area.
2.3.2 Conservation of Momentum
The relationship balances pressure forces with the transport of momentum.
2.3.3 Conservation of Energy
Specific enthalpy is defined by:
Here e is specific internal energy, P pressure, and v = 1/ρ specific volume.
The conservation equations establish the fundamental relationships among pressure, density, velocity, and energy across a shock wave. However, these equations alone do not completely determine the shocked thermodynamic state.
An appropriate equation of state (EOS) is also required to describe how the material responds to compression and changes in internal energy.
In the following lesson, the conservation equations will be used to introduce shock Hugoniots and Rayleigh lines, which provide a graphical representation of the possible states associated with shock compression.