Flow Equations in One Dimension

The introduction of Lagrangian coordinates in the previous section allows a more natural treatment of a continuous flow in one dimension. The derivation of the jump conditions in Section 2.2 made use of a mathematical discontinuity as a simplifying assumption. While this simplification is very useful for many applications, shock waves in reality are not idealized mathematical 

The term shock velocity usually refers to this Eulerian shock velocity relative to the particle velocity of the unshocked material unless otherwise stated, and is usually written as (/ (without the prime). 

Since the Lagrangian walls are impermeable, the mass of the Lagrangian element is constant. At time t, when the walls of the element are separated by an Eulerian distance dx, the density of the fluid within it must be 

This equation can be put in a more useful form by differentiating with respect to time and rearranging 

The compressive force acting per unit area on the mass element from the 

---