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Proof that the Third-Order Time Derivative is Ill Posed

For its tutorial value, let's also show that the PDE of Ruiz [392] 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.$ \,$(C.28), which we restate as

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

yields the characteristic polynomial equation

$\displaystyle 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 [77], it is believed that the finite difference approximation itself provided the damping necessary to eliminate the unstable solution [45]. (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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A Class of Well Posed Damped PDEs