To obtain a restriction about how a solution across a discontinuity propagates we consider the integral conservation law. At first, the conservation law tells us that
where is called density and is called flux. The integral form is
The above equation states that the time rate of change of the total amount of inside the interval must equal the rate that flows into minus the rate that flows out of . Recall that may depend on and through dependence on , then the conservation law can be rewritten as
We now assume that is a smooth curve in spacetime along which suffers a simple discontiuity; that is, assume that is continuously differentiable for and , and that and its derivatives have finite one-sided limits as and . Then choosing and , the integral conservation law may be written
Leibniz’s rule for differentiating an integral whose integrand and limits depend on a parameter can be applied on the left side
Now we take and . The first two terms go to zero because the integrand is bounded and the interval shrinks to zero. Therefore we obtain
where the brackets denote the jump of the quantity inside across the discontinuity (the value of the left minus the value on the right). This is called the jump condition. It relates conditions both ahead of and behind the discontinuity to the speed of discontinuity itself. In this context, the discontinuity in that propagates along the curve is called a shock wave and the curve is called the shock path, or just shock, is called the shock speed and the magnitude of the jump in is called the shock strength.
Source: J.D. Logan, An introduction to nonlinear partial differential equations, 2nd, 2008; Section 3.1.