Proof that the Third-Order Time Derivative is Ill Posed

For its tutorial value, let's also show that the PDE of Ruiz [398] is ill posed, i.e., that at least one component of the solution is a growing exponential. In this case, setting in Eq.(H.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 [80], it is believed that the finite difference approximation itself provided the damping necessary to eliminate the unstable solution [45]. (See §L.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.

---

Order a Hardcopy of Physical Audio Signal Processing

Previous: A Class of Well Posed Damped PDEs
Next: Stability of a Finite Difference Scheme

---

written by Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.