Plotting Lamb Wave Dispersion Curves for Plates

Lamb waves are elastic waves propagating in plate-like structures, with dispersion curves characterizing how wave propagation velocities vary with frequency. To plot Lamb dispersion curves for plates, one must know the material's shear wave velocity and longitudinal wave velocity, parameters that directly influence wave propagation behavior. In code implementation, these material properties are typically defined as input variables, with common approaches using matrix operations to solve the characteristic equations numerically.

The plotting process typically involves solving Lamb wave dispersion equations. These equations describe the relationship between wave frequency and wavenumber, from which phase velocity and group velocity variations with frequency can be derived. For a given material, the shear and longitudinal wave velocities determine the overall shape and characteristics of the dispersion curves. Computational algorithms often employ root-finding methods like the Newton-Raphson iteration or bisection method to solve the transcendental dispersion equations, with frequency ranges discretized into small steps for numerical solutions.

In practical implementation, both symmetric and antisymmetric modes - the two fundamental Lamb wave modes - must be considered. Each mode exhibits different propagation characteristics across various frequency ranges. Dispersion curves help understand the acoustic response of plates at different frequencies, which holds significant importance in nondestructive testing and material characterization applications. Programming implementations typically separate the two modes using different boundary condition formulations, with visualization routines plotting phase velocity (cph) and group velocity (cgr) against frequency-thickness product (fd) on logarithmic or linear scales.