#### 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*:

*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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