The characteristic polynomial equation, from (3.10) with the system matrices given above, in the case of constant coefficients, is

where and are frequency and spatial wavenumber, respectively. There are two pairs of solutions to this equation, which can be written as

and it is simple to show that in contrast with the Euler-Bernoulli beam, the group velocities will be bounded. Indeed, we have in particular that

the first of these relations is similar to that which describes longitudinal wave propagation in a bar, and the second corresponds to shear vibration [77]. For the full varying-coefficient problem, the maximum group velocity, as defined in (3.13), will be