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
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.
Dispersion relations 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.
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 [6], all-angle zero reflection [7, 8], all-direction pulse compression [9] and all-angle zero reflection-zero transmission [10] 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 [9]