Using the WKB method
, we define the [[psi].sub.[mu]] as follows:
Wave propagation has been investigated by matrix algebra method [15,16], WKB method
, and ray method [17-19].
is a method for finding approximate solutions to linear differential equations with spatially varying coefficients.
Using WKB method
, we can further get the following formula:
If the local speed of propagation c(x) slowly varies, asymptotic solutions can be found using the WKB method [12-14].
Maslov, The Complex WKB Method for Nonlinear Equations.
In other words, here we use the same assumptions as in the nonlinear asymptotic WKB method
(Pelinovsky, 1982; Engelbrecht et al., 1988; Didenkulova, 2009).
He begins with the classical WKB method
and proceeds to the theory of Wigner measure, the semi-classical limit from one-dimensional Schrodinger-Poisson to Vlasov-Poisson equations, the semi-classical limits of multidimensional Schrodinger-Poisson equations and the semi-classical limits of the cubic Schrodinger equation in an exterior domain.
(I) Figures 2, 3, and 4 show the potential functions, temporal evolution of the gravitational perturbation, and quasinormal frequency obtained by WKB method in asymptotically flat spacetime.
By making use of (19), we evaluate the QNM frequencies by employing the 6th order WKB method (see Figure 4).
If the local speed of propagation c (x) slowly varies, asymptotic solution scan be found using the WKB method
The sum above represents high order correction in the usual WKB method