By W. D. Jones, H. J. Doucet, J. M. Buzzi (auth.)

ISBN-10: 1475702116

ISBN-13: 9781475702118

ISBN-10: 1475702132

ISBN-13: 9781475702132

Modern plasma physics, encompassing wave-particle interactions and collec tive phenomena attribute of the collision-free nature of scorching plasmas, used to be based in 1946 while 1. D. Landau released his research of linear (small amplitude) waves in such plasmas. It used to be now not till a few ten to 20 years later, even though, with impetus from the then quickly constructing managed fusion box, that enough consciousness used to be committed, in either theoretical and experimental examine, to explain the significance and ramifications of Landau's unique paintings. considering that then, with advances in laboratory, fusion, area, and astrophysical plasma examine, we now have witnessed very important devel opments towards the knowledge of a number of 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 a variety of appli cations. As such, it truly is now often brought in collage physics and engineering curricula on the senior and first-year-graduate degrees. an important prerequisite for all of recent plasma reviews is the less than status oflinear waves in a temporally and spatially dispersive medium comparable to a plasma, together with the kinetic (Landau) conception 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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**Example text**

It is of interest to note that F'(w) is related to F(w) in a simple way. To show this, we first multiply both the numerator and denominator of the righthand side ofEg. 103) by [w - (wo + iv)]. 107) Except for a factor of two, the first term on the right-hand side ofEg. 107) is just F( w). Thus, we see that, by adding an imaginary part to F( w), we have 30 CHAPTER 1 rendered it causal. We leave it for the reader to show that the imaginary part of F' (w) can be expressed as , Im[F (w)] = w-wo 2 2 (w-wo) +v V - ~ f+OO - 00 dw' w-'---w-o-')2:O--+-V"2:-]-(w-'---w-o-) .

This nonphysical result is the consequence of the loss of causality of Eq. (l. 73), and as we will show in the next section is due to the loss of the transient motion in Eq. (1. 73). Our conclusion is that, in contrast to a Fourier transform in time, a Laplace transform in time yields to causal solutions of linear differential equations describing nondissipative systems. 2. Causality and Transient Motion In this section, we will show more precisely how causality is related to the transient motion in a nondissipative system.

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.

### An Introduction to the Linear Theories and Methods of Electrostatic Waves in Plasmas by W. D. Jones, H. J. Doucet, J. M. Buzzi (auth.)

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