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## Characteristics and discontinuities

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 ).
But the latter is only possible when is a characteristic! Otherwise, unique values of u and would determine the unique values of all second derivatives on 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.

Next: Eikonal equation Up: 3: WHY DISCONTINUITIES? Previous: Characteristics
Stanford Exploration Project
1/13/1998