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