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Physical Audio Signal Processing
    Finite Difference Schemes
       Convergence
          Well Posed Initial-Value Problem
             A Class of Well Posed Damped PDEs

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A Class of Well Posed Damped PDEs

A large class of well posed PDEs is given by [45]

$\displaystyle {\ddot y} + 2\sum_{k=0}^M q_k \frac{\partial^{2k+1}y}{\partial x^{2k}\partial t} + \sum_{k=0}^N r_k\frac{\partial^{2k}y}{\partial x^{2k}} \protect$

(N.5)

Thus, to the ideal string wave equation Eq. $\,$ (H.1) we may add any number of even-order partial derivatives in

, plus any number of mixed odd-order partial derivatives in

and

, where differentiation with respect to

occurs only once. Because every member of this class of PDEs is only second-order in time, it is guaranteed to be well posed, as we now show.

To show Eq. $\,$ (N.5) is well posed [45], we must show that the roots of the characteristic polynomial equation have negative real parts, i.e., that they correspond to decaying exponentials instead of growing exponentials. To we insert the general eigensolution

$\displaystyle y(t,x) = e^{st+vx}$

into the PDE just like we did in §H.5 to obtain the so-called characteristic polynomial equation:

$\displaystyle s^2 + 2q(v)s + r(v) = 0$

where

$\begin{eqnarray*} q(v) &\isdef & \sum_{k=0}^M q_k v^{2k}\\ r(v) &\isdef & \sum_{k=0}^N r_k v^{2k} \end{eqnarray*}$

Let's now set , where $k=2\pi/\lambda$ is radian spatial frequency (called the ``wave number'' in acoustics) and of course $j=\sqrt{-1}$ , thereby converting the implicit spatial Laplace transform to a spatial Fourier transform. Since there are only even powers of the spatial Laplace transform variable , the polynomials and are real. Therefore, the roots of the characteristic polynomial equation (the natural frequencies of the time response of the system), are given by

$\displaystyle s = -q \pm \sqrt{q^2 - r}.$

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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.