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
with .
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
where
.
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.