By applying the wavenumber decomposition method , dispersion relations
can be obtained and are illustrated in Figure 6.
The present ACDM includes dispersion relations
[omega](q) (Figure 2) to involve contributions of all atomic vibration frequencies.
The linear form of BBM equation (1) which describes dispersive effect to the equation yields dispersion relation
stating the relation between the wavenumber k and frequency w; the dispersion relation
reads [omega] = k/(1 + [k.sup.2]).
In this case, the deformed energy-momentum dispersion relation
equation (1) will go back to classical one when the energy of the test particle is much lower than [E.sub.p].
By means of the solutions to (5) in both media, one can find the corresponding expressions for [[xi].sub.r], and after merging these solutions, together with those for [p.sub.tot], through appropriate boundary conditions at the interface r = a, one can derive the dispersion relation
of the normal modes propagating in the moving magnetic flux tube.
Considering the nondimensionalized linear dispersion relation
 for deep water with a current, we think that the initial estimation of the nonlinear dispersion relation
Besides, the dispersion relations
of TE waves and TM waves are dual (according to (4) and (8)), so we can get the other dispersion relation
by exchanging the elements of [??] and [??] in either relation ([[micro].sub.x], [[micro].sub.y], and [[epsilon].sub.z] exchange with [[epsilon].sub.x], [[epsilon].sub.y], and [[micro].sub.z]).
where [[beta].sub.2] = [square root of [Q.sub.1]/[rho]2], which is the dispersion relation
of Love-type wave when the interface of the layered half-space is regular.
Slinky-whistler dispersion relation
7a using the linear dispersion relation
(13) are denoted by [[lambda].sub.l], and they are given in Fig.
On the dispersion relation
of random gravity waves, 1, (58) 2 (59)
This phase change can be derived from K-K dispersion relation