#### A Class of Well Posed Damped PDEs

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

Thus, to the ideal string wave equation Eq.(C.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.(D.5) is well posed [45], we must
show that the roots of the characteristic polynomial equation
(§D.3) have negative real parts, *i.e.*, that they correspond to
decaying exponentials instead of growing exponentials. To do this, we
may insert the general eigensolution

*characteristic polynomial equation*:

Let's now set , where
is radian spatial
frequency (called the ``wavenumber'' in acoustics) and of course
, 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

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Proof that the Third-Order Time Derivative is Ill Posed

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Poles at