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.





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



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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Poles at