#### Proof that the Third-Order Time Derivative is Ill Posed

For its tutorial value, let's also show that the PDE of Ruiz  is ill posed, i.e., that at least one component of the solution is a growing exponential. In this case, setting in Eq. (C.28), which we restate as  yields the characteristic polynomial equation By the Routh-Hurwitz theorem, there is at least one root in the right-half -plane. It is interesting to note that Ruiz discovered the exponentially growing solution, but simply dropped it as being non-physical. In the work of Chaigne and Askenfelt , it is believed that the finite difference approximation itself provided the damping necessary to eliminate the unstable solution . (See §7.3.2 for a discussion of how finite difference approximations can introduce damping.) Since the damping effect is sampling-rate dependent, there is an upper bound to the sampling rate that can be used before an unstable mode appears.
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