Let us assume that the solution of the equation (18) has a ``weak discontinuity'', (i.e., discontinuity of the order 2) on the surface . This means that

- and all first derivatives are continuous when crossing the surface .
- satisfies the equation (18) on each side of the surface .
- Some second derivatives have a jump when crossing the surface (they have two different values on both sides of the surface ).

So a characteristic surface is the only possible location for a weak discontinuity
if it is a solution of the equation (18).
If this is valid for a weak discontinuity, it is also valid for
discontinuities of all orders if they represent jumps of derivatives with
respect to *t*. It follows through a simple consideration that: satisfy the same
equation (18), and we can always transform an arbitrary discontinuity
into a weak one.

1/13/1998