Wave Propagation in Linear Media | re-examined

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3Wave propagation in electromagnetic transmission lines After some simple rearrangements, we nally obtain the complete expression for the current along the line I I0 Likewise, we nd the voltage I0 = e ,Xp1, 2 cos T , , 2 Z 1 0 sin X cos T p 1+ 2 1+ 2 , 2 U p L0 =C0 = ,e,Xp 1, 2 1 p sin T + 1= 2 , 1 + 2 Z 1 0 d : p 1+ 2 cos X sin T p 1+ 2 1+ 2 , 2 d : (3.82) (3.83) Before giving numerical evaluations of the above expressions, we take a look at the their range of validity. The lower end of the evanescence region is !0 = 0, i. e. a DC excitation. It can be readily seen by setting =0that this raises no particular problem, and the special case of a non-oscillating step source is covered by our model as we could have expected. Things are di erent for a source oscillating with the plasma frequency !p, which is exactly the boundary between evanescent and propagation mode. The steady-state part of the current (3.69) reduces to the unattenuated oscillation cos !pt, and for t =0,we nd with the identity sign x = 2 R 1 sin x d that the current still complies with the boundary conditions. The 0 voltage, however, does not, because the steady-state solution (3.71) becomes unbounded. The deeper reason for this is our assumption of a lossless plasma, which entails a pole of the characteristic impedance at ! = !p. We could have circumvented this particular problem by choosing a voltage instead of a current source, thereby forcing the current to vanish at resonance. Numerical evaluations of (3.82) and (3.83) are shown in g. 3.13 for the DC case = 0, g. 3.14 for = 0:2 , and g. 3.15 for = 0:8. The most striking property of the result is that the wave front indeed goes straight through the medium with a velocity ofX=T = 1 or, expressed with the unscaled variables, x=t = c. Thus the plasma is completely at rest before the disturbance arrives with the speed of light. This nding is, in principle, nothing spectacular in that it only con rms the more general results obtained by Sommerfeld. It emphasises, however, that even in the evanescent region not the faintest evidence of superluminal wave propagation can be found. Apart from the wave front, there are other similarities between the three cases as well. The in uence of the forced oscillation applied by the signal source diminishes with the attenuation distance 1= 0 = c p 1 , so that further down the line, the wave is determined exclusively !p 1, 2 by the transient solutions (3.74) and (3.76). Note that the transients consist solely of frequency components above the plasma frequency. The evanescent region does not come into play here, which is of course merely a mathematical e ect and bears no physical signi cance. The free oscillations of the plasma far away from both the entrance (as for the current) and 52
3.7 Turn-on e ects in a lossless plasma 30 30 0 20 0 20 10 X 10 X T 10 T 10 20 20 30 1 0 0 0.5 I/I0 0 30 1 0.5 U/U0 Figure 3.13: Evolution of the current (3.82) and voltage (3.83) along a one-dimensional transmission line with a lossless plasma for a non-oscillating step as input current ( = 0). 53 0
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DISSERTATION Wave Propagation in Li
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Kurzfassung Seit der Entdeckung des
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Nullum est iam dictum, quod non sit
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Preface Our popular writers and rep
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the quest for superluminality and t
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Contents Part I Wave propagation ph
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7.3.4 PartitionPoints . . . . . . .
Page 17 and 18: Part I Wave propagation phenomena S
Page 19 and 20: 1.1 Phase and group velocity 1.1 Ph
Page 21 and 22: 1.1 Phase and group velocity ! !c v
Page 23 and 24: 1.2 A few notes on dispersion He co
Page 25 and 26: 1.2 A few notes on dispersion v/c 6
Page 27 and 28: 1.3 Signal velocity dipoles with a
Page 29 and 30: 1.3 Signal velocity The arbitrarine
Page 31 and 32: 1.4 Energy velocity For electromagn
Page 33 and 34: 1.5 Other velocity de nitions For n
Page 35 and 36: 1.5 Other velocity de nitions evide
Page 37 and 38: 2.1 Superluminal wave propagation 2
Page 39 and 40: 2.1 Superluminal wave propagation a
Page 41 and 42: 2.1 Superluminal wave propagation t
Page 43 and 44: 2.2 Quantum mechanical tunnelling e
Page 45 and 46: 2.2 Quantum mechanical tunnelling R
Page 47 and 48: Chapter 3 Wave propagation in elect
Page 49 and 50: 3.1 Model of a transmission line th
Page 51 and 52: 3.2 Excursion: a delay line section
Page 53 and 54: 3.3 Re ection due to termination mi
Page 55 and 56: 3.4 A simple thought experiment I0
Page 57 and 58: 3.5 A dispersive system: the lossle
Page 63 and 64: 3.6 Inhomogeneous transmission line
Page 65 and 66: 3.7 Turn-on e ects in a lossless pl
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Page 77 and 78: 3.8 Turn-on e ects in a wave guide
Page 79 and 80: 3.8 Turn-on e ects in a wave guide
Page 81 and 82: 3.9 A Gaussian pulse in plasma Like
Page 83 and 84: 3.9 A Gaussian pulse in plasma 250
Page 85 and 86: 3.9 A Gaussian pulse in plasma 250
Page 87 and 88: 3.9 A Gaussian pulse in plasma Note
Page 89 and 90: 4.1 The potential step 4.1 The pote
Page 91 and 92: 4.1 The potential step Inside the b
Page 93 and 94: 4.2 Initial wave forms -60 -50 -40
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Page 97 and 98: 4.3 Examples of scattering processe
Page 109 and 110: 4.4 The square barrier they have va
Page 111 and 112: 4.4 The square barrier 1 0.8 0.6 0.
Page 113 and 114: 4.5 Tunnelling time de nitions for
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Page 117 and 118: 4.6 Examples of tunnelling events P
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4.6 Examples of tunnelling events 2
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4.6 Examples of tunnelling events -
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4.6 Examples of tunnelling events t
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4.6 Examples of tunnelling events P
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4.6 Examples of tunnelling events T
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Interlude Wave functions in graphic
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Wave functions in graphical represe
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Part II Numerical aspects of wave e
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5.1 Univariate numerical quadrature
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5.1 Univariate numerical quadrature
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5.2 Convergence acceleration one, t
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5.2 Convergence acceleration (1) ,1
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6.1 Partitioning the integration in
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6.2 Choosing the rst partition poin
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6.3 How to compute the rst integral
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6.3 How to compute the rst integral
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6.4 Asymptotic partition consuming
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6.4 Asymptotic partition of the int
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6.5 Considerations for a Mathematic
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6.6 Controlling the accuracy of the
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Chapter 7 Mathematica implementatio
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7.1 User interface of the function
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7.3 Implementation of OscInt and re
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7.4 Auxiliary functions While[itera
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7.4 Auxiliary functions Linear appr
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7.4 Auxiliary functions For the sak
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7.4 Auxiliary functions Out[8]= 3 f
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7.5 Test of the quadrature routine
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8.1 Preparation of the wave integra
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8.2 Outline of the program structur
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8.2 Outline of the program structur
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8.3 Implementation of the quadratur
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8.3 Implementation of the quadratur
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8.4 Test of the package 8.4 Test of
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8.4 Test of the package -200 -150 -
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8.4 Test of the package 0.4 0.2 -0.
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8.4 Test of the package 8.4.2 Forma
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8.4 Test of the package Out[24]= 0
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A.1 Numerical quadrature Asymptotic
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A.1 Numerical quadrature WynnDegree
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A.1 Numerical quadrature Which[ft =
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A.2 Solutions for the step potentia
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A.3 Solutions for the square barrie
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A.3 Solutions for the square barrie
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A.4 Electromagnetic waves function
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A.5 Utilities for displaying points
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Bibliography Bibliography [1] James
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Bibliography [29] Kurt Edmund Oughs
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Bibliography [59] Ch. Spielmann, R.
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Bibliography [91] C. R. Leavens and
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Bibliography [123] T. O. Espelid an
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Index Index absorption, 7, 9 accura
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Index saddle point integration, 11,
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Curriculum vitae Dipl.-Ing. Thilo S
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