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:
- Asymmetric lamb waves:
With and and . Here denotes the compression wave velocity (longitudinal wave velocity) and is connected via the poission ratio to the shear wave velocity :
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
all roots
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).