Wave Propagation in Linear Media | re-examined

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3Wave propagation in electromagnetic transmission lines which enables one to see that the modulated pulse penetrates deeper into the medium than the base-band signal. This is in accordance with the dispersion relation and the respective attenuation coe cient. The bulk of the signal energy is concentrated within the parabola-shaped region in the timespace graph. But we also recognise a small fraction of the pulse running away down the line when the major part of the signal is already diminishing. The high oscillation frequency of this low but very broad pulse (twice the carrier frequency) shows that it consists of the spectral components just above cuto . In g. 3.24, which shows exactly the spectrum of this particular case, the spectrum seems to be su ciently below cuto , but evidently it is not. We can roughly calculate the frequency that dominates this part of the signal by measuring the slope X= T parallel to the peaks and troughs visible in the graph. They give an estimate for the phase velocity, while the lines connecting the outer ends of the ripples are related to the group velocity. Using the de nition of the dispersion relation (3.97), we readily nd the normalised expressions for the phase velocity, and the group velocity, vp c = r 2 vg c = r 1 , 1 2 2 , 1 ; (3.108) ; (3.109) respectively. With these two equations, which directly correspond to the appropriately measured slope X= T , we nally obtain an interval for the carrier frequency, 2 [1; 1:02]. The left bound of the interval is obtained from the left end of the propagating wave close to X = 0 where the tangents to the ripples are horizontal and the ends of the valleys lie on a vertical line. The same consideration applied to the right end of the wave gives the upper limit of the interval. It is clear, though, that the data we can gather from the graph to nd this limit is directly dependent on the resolution we choose for the voltage amplitude. In reality, the interval is as unlimited as the spectrum of the pulse, and only the numerical accuracy imposes constraints on the reliability of the results (which indeed prevented us from discovering the same e ect in the base-band pulse). One must of course not overlook the fact that the amplitude of the propagating part is nine orders of magnitude smaller than the main pulse. Therefore it is likely to be missed in an experiment. The third example ( g. 3.27) shows a pulse where the carrier frequency equals the cuto frequency ( = 1). Here, half the spectrum lies in the pass band. Thus the pulse is propagated, but it is severely distorted. The reason for this behaviour is evidently the dispersion relation. A moderate pulse distortion requires k(!) to be approximately linear in the frequency range that makes up the dominating part of the spectrum. About cuto , this condition is de - nitely not met, and we end up with a mixture of spectral components having grossly di erent propagation velocities. In g. 3.28, = 1:5 was chosen for the carrier. As we could have expected, this gives a pulse propagating straight through the medium. This result can now be used to verify the 68
3.9 A Gaussian pulse in plasma 250 200 150 100 50 0 0 10 20 30 40 50 Figure 3.27: Evolution of the voltage for = 1. The abscissa denotes X, the ordinate T . 250 200 150 100 50 0 0 10 20 30 40 50 Figure 3.28: Evolution of the voltage for = 1:5. The abscissa denotes X, the ordinate T . 69
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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 . . . . . . .
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Part I Wave propagation phenomena S
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1.1 Phase and group velocity 1.1 Ph
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1.1 Phase and group velocity ! !c v
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1.2 A few notes on dispersion He co
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1.2 A few notes on dispersion v/c 6
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1.3 Signal velocity dipoles with a
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1.3 Signal velocity The arbitrarine
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1.4 Energy velocity For electromagn
Page 33 and 34: 1.5 Other velocity de nitions For n
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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
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
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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
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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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Page 123 and 124: 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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