Eigenvalue problems for the beam and an upper and lower bound for each branch of solutions of Rayleigh-Lamb frequency equation

The Rayleigh-Lamb frequency equation is derived in studying eigenvalues and eigenfunctions for the beam problem in a domain (−a,a) x (−d, d). It is shown that there is an upper bound and a lower bound for each branch of the frequency spectrum which consists of all roots of the frequency equation.