| The studies of plasma wave dispersion law and landau damping are of fundamental interest in plasma physics.The dispersion law for a plasma wave determines the relationship between theangular frequency co and the wave vector k for a wave on the form exp[i(k · r - ωt)]. Ifk is real valued then co may be real valued for un-damped waves, or co may be complex valued, with a positive imaginary part for exponentially growing waves or a negative imaginary part for exponentially decaying waves which called landau damping for longitudinal waves in some case. Dispersion law and Landau damping for the non-relativistic case have been well studied.Advance in laser technology have resulted in a new class of compact, ultra-short pulse lasers with extremely high intensities. Such intensities create novel states of matter, such as relativistic electron; on the other hand, for the high-temperature plasmas, if the thermal energy of the plasma particles encounter at the rest energy of it, both of those case, relativistic plasmas should be studied. The analytical works was presented by Mikhailovskii, but only in the ultra-relativistic limits the discontinuous dispersion laws were given.In this paper, we mainly calculate the dispersion equation numerically and give the full-relativistic dispersion law.This paper is organized as follows:In chapter 1 and 2, the basic concept and dispersion property of plasmas are introduced.In chapter 3, the numerical dispersion law of full-relativistic Longmuir plasmons is given. In the ultra-relativistic, our result is identical to the Mikhailovskii's, and in the weak-relativistic case, the dispersion law returns to the non-relativistic one.In chapter 4, the landau damping of full-relativistic longitudinal plasmon is calculated numerical. In the ultra-relativistic, our result is identical to the Mikhailovskii's one, but only has a small departure.In chapter 5, we mainly discuss the ion-acoustic wave dispersion relationship in...
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