Group velocity of elastic waves as eigensensitivity of a parameter dependent eigenvalue problem

We consider the problem of calculating group velocity of elastic waves along different branches of the dispersion curves without carrying out direct numerical differentiation along these curves. We obtain an expression for the rate of change of eigenvalues with respect to a parameter for a parameter-dependent 'non-linear eigenvalue problem'. Eigenvalues have the meaning of frequency (or its square) and the role of the parameter is played by the wave number. We then define a quotient which shows Rayleigh-quotient-like behaviour in that it differs from unity by a quantity that scales as ∼ϵ2, where ϵ is the magnitude of a parameter used to perturb an eigenvector. By carrying out first order perturbation analysis of a generalised eigenvalue problem involving two Hermitian matrices, at least one of which is positive definite, we obtain an exact expression for group velocity as a rate of change of an eigenvalue. This form is particularly suited for elastic waves. The connection of the eigensensitivity relationship with the stationarity of the Rayleigh's quotient is explored. Two specific examples of mechanical waveguides are presented: (i) the Timoshenko waveguide and (ii) the bending-torsion coupled elastic waveguide.