Wave Propagation in Linear Media | re-examined

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Appendix A Mathematica packages A Mathematica packages The appendix contains source code listings of the most important Mathematica packages that were needed to carry out the numerical evaluation of the wave integrals in the rst part of this work. The last section describes a number of utilities that made life easier with the three-dimensional pictures of propagating waves. A.1 Numerical quadrature The package OscInt was written to compute semi-in nite integrals of strongly oscillating integrands. The user interface and implementation are described in detail in chapter 7. (* Copyright: Copyright 1997, Institute of Computertechnology, *) (* Vienna University of Technology *) (*:Version: Mathematica 2.2.3 *) (*:Title: OscInt *) (*:Revision: 1.2 *) (*:Author: Thilo Sauter *) (*:Keywords: Semi-infinite Quadrature, Very Oscillating Integrands, Series Acceleration *) (*:Requirements: None. *) (*:Warnings: None so far *) (*:Summary: This package provides a set of functions for the numerical quadrature of univariate oscillating integrands over the semi-infinte range {a,Infinity}. The oscillations of the integrand may be irregular and need not show an ultimately constant period. The integrand must be of the form f(x) u(g(x)) with u(x) being Sin[x], Cos[x] or - more generally - Exp[I x]. The argument function g(x) must have an asymptotic expansion at infinity. *) (*:History: 27-12-1996 ZerosInBetween extended to cope with special cases 25-01-1997 PartInt, PartitionTable added, OscInt rewritten 24-03-1997 OscIntControlled, PartitionOffs, PartitionPoints added to allow accuracy control 25-03-1997 Check of the limit of zerof[x] added to functions PartitionTable and PartitionPoints to detect argument functions with finite limit (and thus finite set of zeros) *) BeginPackage["OscInt`"] PolynomialDegree::usage = "PolynomialDegree[f,x] returns the degree of the polynomial asymptotic expansion of f regarding the variable x at Infinity. If f is a pure function or is defined as f[...][x] then x need not be specified." 214
A.1 Numerical quadrature AsymptoticExpand::usage = "AsymptoticExpand[f,x] returns the polynomial part of the asymptotic expansion of f regarding the variable x at Infinity. If f is a pure function or is defined as f[...][x] then x need not be specified." ApproxLimGeneric::usage = "ApproxLimGeneric[f,goal] is used to give a figure of merit for the quality of a polynomial approximation to an argument f of a circular function. It returns the abscissa value where the ratio of the circular velocity of the approximation polynomial and the approximation error, respectively, reaches a given goal (the larger the better). This is equivalent to the number of zeros the approximating function has within one half period of the frequency defined by the difference between f and its approximation." ApproxLimQuad::usage = "ApproxLimHyp[a,b,c,goal] is a special case of ApproxLimGeneric for the function a x^2 + b Sqrt[x^2 + c] + d and its approximation a x^2 + b x + d." ApproxLimLinear::usage = "ApproxLimHyp[a,b,c,goal] is a special case of ApproxLimGeneric for the function a x + b Sqrt[x^2 + c] + d and its approximation (a + b) x + d." QuadZero::usage = "QuadZero[a,b,c,offs][k] returns the k-th root of the quadratic equation a x^2 + b x + c == (k+offs)*Pi for ascending x (the branch to the right of the extremum). The parameter offs has to be determined such that the 0-th solution is already larger than the abscissa value of the extremum (see QuadOffset)." QuadOffset::usage = "QuadOffset[a,b,c] returns the offset value used in QuadZero, which is essentially the ordinate value of the extremum divided by Pi." HypZero::usage = "HypZero[a,b,c,d,offs][k] returns the k-th root of the quadratic equation (a+b) x + d == (k+offs)*Pi, which is the polynomial approximation of a x + b Sqrt[x^2 + c] + d at Infinity for ascending x (the branch to the right of the extremum). The parameter offs has to be determined such that the 0-th solution is already larger than the abscissa value of the extremum (see HypOffset)." HypOffset::usage = "HypOffset[a,b,c,d] returns the offset value used in HypZero, which is essentially the ordinate value of the extremum or bending point divided by Pi." HypZeroExact::usage = "HypZeroExact[a,b,c,d,offs][k] returns the k-th root of the equation a x + b Sqrt[x^2 + c] + d == (k+offs)*Pi for ascending x (the branch to the right of the extremum). The parameter offs has to be determined such that the 0-th solution is already larger than the abscissa value of the extremum (see HypOffsetExact)." HypOffsetExact::usage = "HypOffsetExact[a,b,c,d] returns the offset value used in HypZeroExact, which is in principle the ordinate value of the extremum or bending point divided by Pi." ZerosInBetween::usage = "ZerosInBetween[f,a,b] returns the number of solutions of the function g[x] == k*Pi for x in (a,b). The function f[k] is required to give the k-th solution of g[x] == k*Pi. If f[0] lies in the interval (a,b) only the part to the right of f[0] is regarded. With this function, the parameter NSumTerms of NSum can be determined." HypApproxError::usage = "HypApproxError[b,c,goal] returns the abscissa value where the error between the function b*Sqrt[x^2 + c] and its approximation b*x reaches a given value." PartInt::usage = "PartInt[f,partpoints,a,(opts)] computes the indefinite integral of a function f over the range (a,Infinity) using a partition-extrapolation method when the partition points partpoints are explicitly given. The first interval (a,p1) is obtained using the double-exponential method, the others by a Gauss-Kronrod formula. The partial integrals between the partition points are summed up and passed to SequenceLimit to determine the value of the integral. If the first member of the list of partition points in Infinity, the no extrapolation is performed, and the integral is entirely calculated with the double-exponential rule. Note that the partition points must be of ascending order and chosen such that extrapolation is possible (in case of an oscillating integrand, they must be to the right of the rightmost extremum or saddle point of the argument function)." PartitionTable::usage = "PartitionTable[f,a,n,(opts)] returns a list of partition points suitable for PartInt. Depending on the option FunctionType the function f is interpreted either as argument function of Sin[f[x]] or as a function giving the k-th zero of Sin[arg[x]]. In the first case, the first n solutions of f[x] == K*Pi to the right of the lower 215
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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
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1.5 Other velocity de nitions For n
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1.5 Other velocity de nitions evide
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2.1 Superluminal wave propagation 2
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2.1 Superluminal wave propagation a
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2.1 Superluminal wave propagation t
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2.2 Quantum mechanical tunnelling e
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2.2 Quantum mechanical tunnelling R
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Chapter 3 Wave propagation in elect
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3.1 Model of a transmission line th
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3.2 Excursion: a delay line section
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3.3 Re ection due to termination mi
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3.4 A simple thought experiment I0
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3.5 A dispersive system: the lossle
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3.6 Inhomogeneous transmission line
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3.7 Turn-on e ects in a lossless pl
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3.8 Turn-on e ects in a wave guide
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3.8 Turn-on e ects in a wave guide
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3.9 A Gaussian pulse in plasma Like
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3.9 A Gaussian pulse in plasma 250
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3.9 A Gaussian pulse in plasma 250
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3.9 A Gaussian pulse in plasma Note
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4.1 The potential step 4.1 The pote
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4.1 The potential step Inside the b
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4.2 Initial wave forms -60 -50 -40
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4.2 Initial wave forms -60 -50 -40
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4.3 Examples of scattering processe
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4.4 The square barrier they have va
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4.4 The square barrier 1 0.8 0.6 0.
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4.5 Tunnelling time de nitions for
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4.5 Tunnelling time de nitions for
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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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Page 181 and 182: 7.3 Implementation of OscInt and re
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Page 207 and 208: 8.1 Preparation of the wave integra
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Page 217 and 218: 8.3 Implementation of the quadratur
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Page 221 and 222: 8.4 Test of the package 8.4 Test of
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Page 227 and 228: 8.4 Test of the package 8.4.2 Forma
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Page 233 and 234: A.1 Numerical quadrature WynnDegree
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Page 237 and 238: A.2 Solutions for the step potentia
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Page 249 and 250: A.4 Electromagnetic waves function
Page 251 and 252: A.5 Utilities for displaying points
Page 253 and 254: Bibliography Bibliography [1] James
Page 255 and 256: Bibliography [29] Kurt Edmund Oughs
Page 257 and 258: Bibliography [59] Ch. Spielmann, R.
Page 259 and 260: Bibliography [91] C. R. Leavens and
Page 261 and 262: Bibliography [123] T. O. Espelid an
Page 263 and 264: Index Index absorption, 7, 9 accura
Page 265 and 266: Index saddle point integration, 11,
Page 267: Curriculum vitae Dipl.-Ing. Thilo S
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