A Class of Well Posed Damped PDEs
A large class of well posed
PDEs is given by [
45]

 |
(D.5) |
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
into the PDE just like we did in §
C.5 to obtain the so-called
characteristic polynomial equation:
where
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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