English: Free plate wave (Lamb wave) dispersion curves (symmetric S and asymmetric A) for two different poisson's ratios (sigma=0,27 e.g. steel and sigma 0,34 e.g. titan and aluminium)

• The y axis shows the phase velocity v normalized to the shear velocity v_s
• The x axis shows the product of angular frequency omega and plate thickness d normalized to the shear velocity v_s

To reproduce these curves find the roots of the following 2 equations:

• Symmetric lamb waves: ${\displaystyle \tan(\alpha d/2)\cot(\beta d/2)=-p^{2}/\alpha \beta }$
• Asymmetric lamb waves:${\displaystyle \tan(\alpha d/2)\cot(\beta d/2)=-\alpha \beta /p^{2}}$

With ${\displaystyle \alpha ^{2}=\omega ^{2}/v_{l}^{2}-k^{2}}$ and ${\displaystyle \beta ^{2}=\omega ^{2}/v_{s}^{2}-k^{2}}$ and ${\displaystyle p^{2}=(k^{2}-(\omega ^{2}/v_{s}^{2})/2)^{2}k^{-2}}$. Here ${\displaystyle v_{l}}$ denotes the compression wave velocity (longitudinal wave velocity) and is connected via the poission ratio ${\displaystyle \sigma }$ to the shear wave velocity ${\displaystyle v_{s}}$: ${\displaystyle v_{s}/v_{l}={\sqrt {(1-2\sigma )/(2-2\sigma )}}.}$

Please refer to the following books:

• Brekhovskikh: Waves in layered media in Applied Mathematics and Mechanics, 16 (1980), p. 60ff
• Joseph L. Rose: Ultrasonic Waves in Solid Media, Cambridge University Press, 2004, p. 110

It is not straight forward to compute these curves. These curves where calculate with help of the pyhton-package scipy.optimize and especially the function fsolve and the numerical continuation tool AUTO07p.

For a certain value (here 20) of ${\displaystyle \omega d/v_{s}}$ all roots ${\displaystyle v/v_{s}}$ were computed with fsolve. These ”starting roots“ then were plugged in AUTO07p and the tool ”continued“ the branches to the left. All branches except the S0 and A0 end in a limit point (these modes have ”cut off“ frequencies).