# Rayleigh-Lamb dispersion curves

In an infinite plane elastic plate of thickness $d$, it is shown that the modes of oscillation corresponding to a fixed time-frequency $\omega$ have wave-numbers given by solutions of the Rayleigh-Lamb equations

$$\frac{\tan (pd)}{\tan (qd)}=-\left[\frac{4k^2pq}{\left(k^2-q^2\right)^2}\right]^{\pm 1}$$

where the +1 exponent corresponds to symmetric modes and the -1 exponent to antisymmetric modes, and

$$p^2=\frac{\omega ^2}{c_L^2}-k^2$$

$$q^2=\frac{\omega ^2}{c_T^2}-k^2$$

where $c_L^2=\frac{2\mu +\lambda }{\rho }$ is the longitudinal wave-speed and $c_T^2=\frac{\mu }{\rho }$ is the transverse wave-speed ($\mu$ and $\lambda$ being the Lame constants). From this equation one obtains certain dispersion curves, relating the wave-number of each mode to the frequency. My question is: For each value of the frequency, the Rayliegh-Lamb equations give a discrete number of wave-numbers. How do you decide which wave-number belongs to which mode? That is, if I have a set of wave-numbers corresponding to one value of the frequency, say $\omega_1$, and then I get a new set of wave-numbers corresponding to a different frequency, say $\omega_2$, then how do I identify wave-numbers corresponding to different frequencies as belonging to the same mode?

I hope I've explained myself adequately. You can see that my question can also be framed for many other trascendental equations with multiple roots that show up in mathematical physics. I chose this particular problem to make my point. One might think that it is just a matter of ordering the wave-numbers in increasing order. But I've seen dispersion curves in which the curves for different modes intersect, so an ordering criterion is not applicable (not sure if this happens for the Rayleigh-Lamb modes, though)

Thanks. And sorry for the brick of text.

• Good question. This isn't my area of expertise so I don't know anything about it, but I hope you get a good answer. – David Z Jun 8 '11 at 0:25

Let me see if I understand the question correctly: in general our solutions for a system will involve for every wavevector k a set of frequencies $\omega(k)$. When these curves giving $\omega(k)$ do not cross, there is in obvious sense in which we can separate our solutions into different modes. But when they do cross how do we assign modes?