## Characteristic Polynomial Equation

The *characteristic polynomial equation* for a linear PDE with
constant coefficients is obtained by taking the 2D Laplace transform
of the PDE with respect to and . A simple way of doing this is
to substitute the general eigensolution

into the PDE, where denotes the complex variable associated with the Laplace-transform with respect to time, and is the complex variable associated with the

*spatial*Laplace transform.

As a simple example, the ideal-string wave equation (analyzed in §C.1) is a simple second-order PDE given by

where is a positive constant (sound speed, as discussed in §C.1).

Substituting Eq.(D.6) into Eq.(D.7) results in the following
*characteristic polynomial equation*:

*dispersion relation*:

*i.e.*, using Fourier transforms in place of Laplace transforms),

*phase velocity*of a traveling wave is, by definition, the temporal frequency divided by spatial frequency, we have simply

*dispersive*, in which case the phase velocity depends on frequency (see §C.6 for an analysis of stiff vibrating strings, which are dispersive). Moreover, wave propagation may be

*damped*in a frequency-dependent way, in which case one or more roots of the characteristic polynomial equation will have negative real parts; if any roots have positive real parts, we say the initial-value problem is

*ill posed*since is has exponentially growing solutions in response to initial conditions.

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