Chapter 3
Conditions Across the Shock: The
Rankine-Hugoniot Equations
3.1 Introduction to Normal Shock Waves
In this chapter we will investigate the relationship between the states on both sides of
a normal shock wave. These relationships are known as the Rankine-Hugoniot
equations and they can be derived by applying the laws of mass, momentum and
energy conservation. The relationships derived will be used in the subsequent
chapters and, in particular, they will be used to ascertain the accuracy of the solutions
obtained numerically.
3.2 Conservation Equations
The laws of mass, momentum and energy conservation are applied to a fluid
traversing the shock front [1–6]. Let us consider a shock wave propagating to the
right into a stationary fluid with velocity U s as shown in Fig. 3.1a. The pressure and
density of the fluid ahead of the shock front are assumed to be p 0 and ρ 0 , respectively, while the compressed fluid behind the shock front is moving with velocity u p
and has pressure p and density ρ. In a reference system in which the shock is
stationary as illustrated in Fig. 3.1b, the velocity of the fluid entering the shock is
U s and the velocity of the fluid leaving the shock is U s À u p .
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021
S. Prunty, Introduction to Simple Shock Waves in Air, Shock Wave and High
Pressure Phenomena, https://doi.org/10.1007/978-3-030-63606-7_3
89
Conditions Across the Shock: The
Rankine-Hugoniot Equations
3.1 Introduction to Normal Shock Waves
In this chapter we will investigate the relationship between the states on both sides of
a normal shock wave. These relationships are known as the Rankine-Hugoniot
equations and they can be derived by applying the laws of mass, momentum and
energy conservation. The relationships derived will be used in the subsequent
chapters and, in particular, they will be used to ascertain the accuracy of the solutions
obtained numerically.
3.2 Conservation Equations
The laws of mass, momentum and energy conservation are applied to a fluid
traversing the shock front [1–6]. Let us consider a shock wave propagating to the
right into a stationary fluid with velocity U s as shown in Fig. 3.1a. The pressure and
density of the fluid ahead of the shock front are assumed to be p 0 and ρ 0 , respectively, while the compressed fluid behind the shock front is moving with velocity u p
and has pressure p and density ρ. In a reference system in which the shock is
stationary as illustrated in Fig. 3.1b, the velocity of the fluid entering the shock is
U s and the velocity of the fluid leaving the shock is U s À u p .
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021
S. Prunty, Introduction to Simple Shock Waves in Air, Shock Wave and High
Pressure Phenomena, https://doi.org/10.1007/978-3-030-63606-7_3
89