This change of variables transforms propagative modes to evanescent ones if [beta] 54 0.
2], it is easy to show that for propagative modes [[LAMBDA].
For the artificial boundary conditions to be most effective, they need to be able to absorb not only the propagative modes of the solution, but also the evanescent ones.
This limit is due to difference between the true velocity which was used to construct the PML discretization and the f-d velocity in the propagative part, and decreases from 0.
The most difficult and the most important for the finitedifference approximation of wave problems are the propagative modes, that is, those with purely imaginary exponents.
we see that these propagative modes correspond to [[eta].
Consider now the modes which are propagative in x, that is, when c < [a.
k] by considering the following form of a propagative mode:
So, the error for the Dirichlet data corresponding to the propagative modes will depend on
2], 0] corresponds to the propagative modes, and the curvilinear part corresponds to the evanescent modes.