Dear Arindam Nath, what is the pair (c,k)?

c - the sound speed velocity and the k - wavelenght?

c - the elastic constant of a shear elastic wave SH and the k - wavelength?

In the first case, the literature uses the basic equation v = f λ, with f frequency and λ the wavelength, then it becomes clear how to do it, with the 2 π factor, i.e., the face velocity of a p/sound wave c = v = ω/k (please do not confuse with the cp at constant pressure or cs adiabatic sound speeds used in thermodynamics for fluids and solids).

In the second case which I guess is yours, the secular equation is the Christoffel equation for solids which has 3 solutions, for P, SV, and SH elastic waves:

κ γ c γT u = ρ ω2 u (1)

Equation (1) from: https://wiki.seg.org/wiki/Dictionary:Christoffel_equation/es

where is κ the waves number, γ is the direction of the cosine matrix, c is the stress tensor, u the displacement vector of the particle or polarization, ρ the solid density, and ω the angular frequency.

To add attenuation in the second case for an SH wave, follow the paper:

Yaping Zhu and Ilya Tsvankin, (2006), "Plane-wave propagation in attenuative transversely isotropic media," GEOPHYSICS 71: T17-T30.

Article Plane-wave propagation in attenuative transversely isotropic media

This paper develops a procedure for a VTI media, i.e., the same as hexagonal point symmetry case. They plot the attenuation as a function of a Q factor which is related to the constant stiffness c & k. The plot is there and a procedure description. They emphasize the SH wave case on page 19. The plot is given as figure 1 for the SH case on page 20.

Best Regards.

While plotting the phase velocity, we took a real part and while attenuation, we take imagniary part. If you are unable to separate it. give the command to the imaginary part.

ContourPlot[Im[exp1] == 0 , {x, 0.1, 30}, {c, 5200, 6000}]

Please, have a look at this paper:

https://doi.org/10.1016/j.mechrescom.2016.03.007

Here, Legendre polynomial and Newton Raphson methods were selected.

Dear Arindam,

I suppose that you have already obtained the modal solutions as complex wavenumbers k. While the phase velocity cp is given by

cp = w/real(k),

where w is the angular frequency (= 2 pi f) and real(k) is the real part of the wavenumber, the attenuation a is given by

a = 2 pi imag(k)/real(k),

where imag(k) is the imaginary part of the wavenumber. The unit of a is Nepers per wavelength.

The attenuation can be depicted as the rate of decrease in amplitude! Two different approach can be used, either take wave number complex or phase velocity as complex, then fix the real part and plot the imaginary part vs wave number or phase velocity