By W. D. Jones, H. J. Doucet, J. M. Buzzi (auth.)
Modern plasma physics, encompassing wave-particle interactions and collec tive phenomena attribute of the collision-free nature of scorching plasmas, was once based in 1946 whilst 1. D. Landau released his research of linear (small amplitude) waves in such plasmas. It used to be now not until eventually a few ten to 20 years later, in spite of the fact that, with impetus from the then quickly constructing managed fusion box, that adequate awareness used to be committed, in either theoretical and experimental study, to explain the significance and ramifications of Landau's unique paintings. given that then, with advances in laboratory, fusion, house, and astrophysical plasma study, we've got witnessed very important devel opments towards the certainty of numerous linear in addition to nonlinear plasma phenomena, together with plasma turbulence. at the present time, plasma physics stands as a well-developed self-discipline containing a unified physique of robust theoretical and experimental ideas and together with quite a lot of appli cations. As such, it really is now usually brought in college physics and engineering curricula on the senior and first-year-graduate degrees. an important prerequisite for all of contemporary plasma reports is the lower than status oflinear waves in a temporally and spatially dispersive medium akin to a plasma, together with the kinetic (Landau) idea description of such waves. instructing adventure has frequently proven that scholars (seniors and first-year graduates), while first uncovered to the kinetic thought of plasma waves, have problems in facing the necessary sophistication in multidimensional advanced variable (singular) integrals and transforms.
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Additional info for An Introduction to the Linear Theories and Methods of Electrostatic Waves in Plasmas
That is, the result violates causality. If Wo (the frequency of maximum transmissivity of the filter) is chosen to correspond to red light, the theoretical result predicts that red light should be seen coming through the filter before the white-light pulse is produced. This absurdity is sometimes referred to as the red-filter paradox. The purpose of the preceding exercise was to show the importance of the Kramers-Kronig relations in connection with causality. We leave it as an exercise for the reader to show that F(w) does not satisfy the Kramers-Kronig relations.
Let us first consider the case where t < O. 55 ) since all the poles of F(w) are outside the contour C [see Fig. 4(a)]. Also, --+ 00 [see Eq. 50)] we may apply Jordan's lemma since F( w) --+ 0 for (see Appendix A) to J2(t,R) to conclude that Iwl lim 12(t < 0, R) = O. 56) 00 we can write 00) = I1(t < O,R --+ 00) = f +oo+;e - 00 . F(w)e-uDtdw = O. 57) Therefore, comparing Eq. 57) with Eq. 52) we can immediately conclude that M(k, t < 0) = O. Thus, as already anticipated, we see that indeed our 18 CHAPTER 1 Laplace transform has generated a causal solution: nothing happens before the current in the antenna is turned on (at t = 0, by definition).
Doing this, we find that Iwol, 1 t R = - 2ni [ ] (, ) e-iwot (wo + O)(wo - 0) + e- iQt (0 + w o)20 + eiQt] (0 - w o)20 . 60) Also, as before, since F(w) ~ 0 for to (t,R) to show that Y Iwl ~ 00, we may apply Jordan's lemma lim Y(t > O,R) R-oo Therefore, for R = o. 61 ) ~ 00 ](t>O,R~oo)=Y(t,R~oo)= f+ oo + ie . F(w)e-1wtdw. 62) Thus, comparing Eq. 62) with Eq. 52) and using the result ofEq. 60), we can write for t > 0 that r:t. 63) The solution obtained is identical to the solution M(t) given by Eq.
An Introduction to the Linear Theories and Methods of Electrostatic Waves in Plasmas by W. D. Jones, H. J. Doucet, J. M. Buzzi (auth.)