Proof that the Third-Order Time Derivative is Ill Posed

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Physical Audio Signal Processing

Well Posed Initial-Value Problem

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The sampling rate of a discrete-time signal is defined as the number of samples per second. Its units are thus in Hertz (Hz). Shannon's Sampling Theorem states that the original continuous-time signal can be recovered exactly from the samples if and only if the sampling rate is higher than twice the highest frequency present in the original signal. Any higher frequencies will alias to frequencies below half the sampling rate. — Click for https://ccrma.stanford.edu/~jos/mdft/Sampling_Theory.html

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An exponential is any function of the form Ae^-t/τ, where t is the independent variable (often denoting time in seconds), A is an arbitrary scale factor, τ is called the time constant, and e is Euler's number (2.718281828459...). Many systems in nature exhibit exponential decay over time, because an exponential decay is produced whenever the rate of decay is proportional to how much is left. — Click for https://ccrma.stanford.edu/~jos/mdft/Exponentials.html

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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 [395] 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

-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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``Physical Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2010, ISBN 978-0-9745607-2-4
Copyright © 2024-06-28 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

Proof that the Third-Order Time Derivative is Ill Posed

``Physical Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2010, ISBN 978-0-9745607-2-4 Copyright © 2024-06-28 by Julius O. Smith III Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

``Physical Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2010, ISBN 978-0-9745607-2-4
Copyright © 2024-06-28 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA), Stanford University