# What are the phase speeds of $A_0$ and $S_0$ Lamb waves at very low frequencies?

Lamb waves are elastic wave modes in plates. In general, they are mathematically complex to deal with, and numerical solutions are typically used to find dispersion curves for each mode, as shown in this image from the Wikipedia article [image by the user Svebert, licensed as CC BY 3.0]:

If I have understood things correctly, it is not practically possible to find analytical expressions for every Lamb wave mode. Still, we do know some things about the behaviour of the $$A_0$$ and $$S_0$$ modes in the high- and low-frequency limits:

• At high frequencies, the phase speed for $$A_0$$ and $$S_0$$ both converge to the Rayleigh wave speed.
• At low frequencies, the $$A_0$$ phase speed is approximately proportional to the square root of the frequency.
• At low frequencies, the $$S_0$$ phase speed is approximately independent of frequency.

My question is thus: What are the approximate analytical expressions for the phase speed of $$A_0$$ and $$S_0$$ Lamb waves at low frequencies, as a function of the plate's linear elastic parameters?

At low frequencies, the Lamb phase velocities of the $$A_{0}$$ and $$S_{0}$$ modes are consistent with those predicted by plate theories from structural mechanics. For the Kirchoff-Love plate theory, the velocity of the $$A_{0}$$ mode may be found from the equation decribing the flexural behaviour, as: $$v_{flexion}=\left( \frac{D}{\rho h} \right)^{1/4}\sqrt{\omega}$$ where $$D= \frac{Eh^{3}}{12(1-\nu)^2}$$ is the plate bending stiffness or plate flexural rigidity. For the compression ($$S_{0}$$) mode, the low frequency limit is simply: $$v_{compresssion}=\sqrt{\frac{E}{(1-\nu^{2})\rho} }$$ These expressions may also be found using Lamb dispersion equation and letting $$k, \omega \to 0$$.