By expressing the Lagrangian displacements [[xi].sub.ir] and [[xi].sub.er] in both media via the derivatives of corresponding Bessel functions and by applying the boundary conditions for continuity of the pressure perturbation [p.sub.tot] and [[xi].sub.r] across the interface, r = a, one obtains the dispersion relation
of normal MHD modes propagating in a flowing compressible jet surrounded by a static compressible plasma [65, 83, 84]
For Figure 1, with wave vectors [[bar.k].sup.i.sup.TE] = [??][k.sup.TE.sub.0x] + [??][k.sup.TE.sub.y] (incident) and [[bar.k].sub.r.sup.TE] = [??][k.sup.TE.sub.0x] + [??][k.sup.TE.sub.y] (reflection) in the isotropic medium and [[bar.k].sub.t.sup.TE] = [??][k.sup.TE.sub.tx] (transmission) in the biaxial anisotropic medium, the dispersion relations
can be expressed as
The dispersion relation
of Love-type wave has been obtained and coincides with the classical dispersion relation
of Love wave in particular cases.
Although the number of the data is limited, we are able to say that the linear dispersion relation
can be used ordinarily for the ocean waves except for the special cases where very steep waves are generated by extremely strong wind.
have been obtained separately in the cases of electrical open circuit and short circuit.
We have used the expressions of [eta], [[phi].sup.(1)], [[phi].sup.(2)], [[psi].sup.(1)], [[psi].sup.(2)], and -[p.sup.v.sub.1] + [p.sup.v.sub.2] in (34) to find the dispersion relation
which is a quadratic equation expressed as follows:
The condition of nontrivial solution of the above system gives the following biquadratic dispersion relation
It is not easy to achieve fast electrokinetic mode in presence of drifting carriers in the medium; hence, we will study this dispersion relation
under slow electrokinetic mode situation only.
In order to obtain the propagation constants for the EH0i , [EH.sub.11], and [EH.sub.-11] modes, we solve the dispersion relation
as stated in (7) for the range of [beta].
This results in an unusual hyperbolic dispersion relation
for the medium and consequently many interesting phenomena such as hyperlensing [2-5], control of the electromagnetic fields , all-angle zero reflection [7, 8], all-direction pulse compression  and all-angle zero reflection-zero transmission  can be happened.
In particular, the exact dispersion relation
of the plasma column loaded cylindrical waveguide was presented in different forms with different notations in [15, 48, 51, 52].
The dispersion relation
for TM waves in the infinite periodic structure, which relates the frequency [omega], the longitudinal wave number [k.sub.x] and the Bloch wave number [bar.k], can be written as