Characteristics and discontinuities

Let us assume that the solution $u({\bf r},t)$of the equation (18) has a ``weak discontinuity'', (i.e., discontinuity of the order 2) on the surface $\omega ({\bf r},t)=0$. This means that

• $u({\bf r},t)$ and all first derivatives are continuous when crossing the surface $\omega$.
• $u({\bf r},t)$ satisfies the equation (18) on each side of the surface $\omega$.
• Some second derivatives have a jump when crossing the surface $\omega$(they have two different values on both sides of the surface $\omega$).

But the latter is only possible when $\omega$ is a characteristic! Otherwise, unique values of u and ${\partial u\over \partial x_{i}}$ would determine the unique values of all second derivatives on the surface $\omega$.

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: $u({\bf r},t), \int_{0}^{t} u({\bf r},t)dt, \dot{u}({\bf r},t),\ddot{u}({\bf r},t), \ldots$ satisfy the same equation (18), and we can always transform an arbitrary discontinuity into a weak one.