Proof that the Third-Order Time Derivative is Ill Posed

For its tutorial value, let's also show that the PDE of Ruiz [365] is ill posed, i.e., that at least one component of the solution is a growing exponential. In this case, setting $y(t,x) = e^{st+jkx}$ in Eq. (G.28), which we restate as

$$Ky''= \epsilon {\ddot y}+ \mu{\dot y}+ \mu_3{\dddot y},$$

yields the characteristic polynomial equation

$$p(s,jk) = \mu_3 s^3 + \epsilon s^2 + \mu s + Kk^2 = 0.$$

By the Routh-Hurwitz theorem, there is at least one root in the right-half $s$-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 [74], it is believed that the finite difference approximation itself provided the damping necessary to eliminate the unstable solution [43]. (See §J.4.1 for an 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.