LAPLACIAN EIGENSTRUCTURE OF THE EQUILATERAL TRIANGLE Brian J. McCartin Applied Mathematics Kettering University HIKARI LT D HIKARI LTD Hikari Ltd is a publisher of international scientific journals and books. www.m-hikari.com Brian J. McCartin, LAPLACIAN EIGENSTRUCTURE OF THE EQUI- LATERAL TRIANGLE, First published 2011. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, without the prior permission of the publisher Hikari Ltd. ISBN 978-954-91999-6-3 Copyright c 2011 by Brian J. McCartin Typeset using LATEX. Mathematics Subject Classification: 34B24, 34L05, 34M25, 35C05, 35J05, 35P05, 35P10 Keywords: Laplacian eigenvalues/eigenvectors, equilateral triangle, Dirich- let problem, Neumann problem, Robin problem, radiation boundary condi- tion, absorbing boundary condition, impedance boundary condition, polygonal domains, trigonometric eigenfunctions, cylindrical waveguides, modal degen- eracy, Eisenstein primes, Sturm-Liouville boundary value problem, non-self- adjoint boundary value problem Published by Hikari Ltd Dedicated to my soul-mate Barbara Ann (Rowe) McCartin for making Life worth living. Lam´e’s pi`ece de r´esistance [40] Preface v PREFACE Mathematical analysis of problems of diffusion and wave propagation fre- quently requires knowledge of the eigenvalues and eigenfunctions of the Lapla- cian on two-dimensional domains under various boundary conditions. For gen- eral regions, this eigenstructure must be numerically approximated. However, for certain simple shapes the eigenstructure of the Laplacian is known ana- lytically. (An advantage of analytical expressions for the eigenstructure over numerical approximations is that they permit parametric differentiation and, consequently, sensitivity and optimization studies.) The simplest and most widely known such domain is the rectangle. Being the cartesian product of two intervals, its eigenstructure is expressible in terms of the corresponding one-dimensional eigenstructure which in turn is comprised of sines and cosines. Not nearly as well known, in 1833, Gabriel Lam´ediscovered analytical formulae for the complete eigenstructure of the Laplacian on the equilateral triangle under either Dirichlet or Neumann boundary conditions and a portion of the corresponding eigenstructure under a Robin boundary condition. Sur- prisingly, the associated eigenfunctions are also trigonometric. The physical context for his pioneering investigation was the propagation of heat throughout polyhedral bodies. For the better part of the last decade, the present author has sought to explicate, extend and apply these ingenious results of Lam´e. The present book narrates this mathematical journey by providing a com- plete and self-contained treatment of the eigenstructure of the Laplacian on an equilateral triangle. The historical context and practical significance of the problem is carefully traced. The separate cases of Dirichlet, Neumann, radiation, absorbing and impedance boundary conditions are individually and exhaustively treated with the Dirichlet and Neumann cases also extended from the continuous to the discrete Laplacian. Corresponding results for the Sturm- Liouville boundary value problem under an impedance boundary condition are reviewed and applied to the parallel plate waveguide. Polygons with trigono- metric eigenfunctions receive comprehensive study. Application to modal de- generacy in equilateral triangular waveguides has also been included. Mysteries of the Equilateral Triangle [62] surveys the mathematical proper- ties of the equilateral triangle while Lore & Lure of the Laplacian [63] explores the same for the Laplacian. Consequently, Chapter 1 commences with but a brief r´esum´eof those facets of the equilateral triangle and the Laplacian which are most germane to the present study. Chapters 2 and 3 present Lam´e’s analysis of the eigenstructure of the Laplacian on an equilateral triangle under Dirichlet and Neumann boundary conditions, respectively. Chapter 4 develops a complete classification of those polygonal domains possessing trigonometric eigenfunctions under these same boundary conditions while Chapter 5 applies vi Preface Lam´e’sanalysis to the investigation of modal degeneracy in acoustic and elec- tromagnetic waveguides of equilateral triangular cross-section. In the remaining chapters, Lam´e’sanalysis is extended to Robin bound- ary conditions. Chapter 6 considers the radiation boundary condition while Chapter 7 is devoted to the absorbing boundary condition. The next two chap- ters are devoted to the important practical case of the impedance boundary condition. Chapter 8 reviews the case of the one-dimensional analysis of the Sturm-Liouville boundary value problem and Chapter 9 then generalizes this analysis to the case of the two-dimensional equilateral triangle. Chapter 10 surveys some alternative approaches to the equilateral triangular eigenprob- lem from the literature. Finally, Appendix A presents the extension of the analysis of the eigenstructure of the equilateral triangle from the continuous to the discrete Laplacian with Dirichlet or Neumann boundary conditions. Some moderate redundancy has been incorporated into the presentation so as to endow each chapter with a modicum of independence. Throughout the exposition, enough background material is provided so as to make this mono- graph accessible to a wide scientific audience. Unless otherwise attributed, the source material for the biographical vignettes sprinkled throughout the text was drawn from Biographical Dictionary of Mathematicians [23], MacTutor History of Mathematics [71] and Wikipedia, The Free Encyclopedia [91]. The target audience for the book consists of practicing Engineers, Scien- tists and Applied Mathematicians. Particular emphasis has been placed upon including sufficient prerequisites to make the book accessible to graduate stu- dents in these same fields. In point of fact, the bulk of the subject matter has been developed at a mathematical level that should be accessible to advanced undergraduates studying Applied Mathematics. The goal of the book has been not only to provide its readership with an understanding of the theory but also to give an appreciation for the context of this problem within the corpus of Applied Mathematics as well as to include sufficient applications for them to apply the results in their own work. I owe a debt of gratitude to a succession of highly professional Interlibrary Loan Coordinators at Kettering University: Joyce Keys, Meg Wickman and Bruce Deitz. Quite frankly, without their tireless efforts in tracking down many times sketchy citations, whatever scholarly value may be attached to the present work would be substantially diminished. Also, I would like to warmly thank my Professors: Oved Shisha, Ghasi Verma and Antony Jameson. Each of them has played a significant role in my mathematical development and for that I am truly grateful. As always, my loving wife Barbara A. (Rowe) McCartin bears responsibility for the high quality of the mathematical illustrations. Brian J. McCartin Fellow of the Electromagnetics Academy Editorial Board, Applied Mathematical Sciences Contents Preface .................................. vi 1 Prolegomenon 1 1.1 The Equilateral Triangle . 2 1.1.1 Triangular Coordinates . 2 1.1.2 Symmetric-Antisymmetric Decomposition . 3 1.2 The Laplacian . 4 1.2.1 Eigenstructure . 4 1.2.2 Separation of Variables . 5 1.3 Gabriel Lam´e . 6 2 The Dirichlet Problem 8 2.1 The Dirichlet Eigenproblem for the Equilateral Triangle . 8 2.2 Lam´e’sFundamental Theorem I . 9 2.3 Construction of Modes . 12 2.3.1 Symmetric Modes . 12 2.3.2 Antisymmetric Modes . 15 2.4 Modal Properties . 16 2.4.1 Orthonormality . 18 2.4.2 Completeness . 19 2.4.3 Nodal Lines . 21 2.5 Spectral Properties . 24 2.6 Green’s Function . 25 2.7 Related Structures . 26 2.7.1 Hemiequilateral Triangle . 26 2.7.2 Regular Rhombus . 26 2.7.3 Regular Hexagon . 27 2.8 Peter Gustav Lejeune Dirichlet . 28 3 The Neumann Problem 30 3.1 The Neumann Eigenproblem for the Equilateral Triangle . 30 3.2 Lam´e’sFundamental Theorem II . 31 3.3 Construction of Modes . 33 vii viii Preface 3.3.1 Symmetric Modes . 33 3.3.2 Antisymmetric Modes . 37 3.4 Modal Properties . 38 3.4.1 Orthonormality . 41 3.4.2 Completeness . 42 3.4.3 Nodal / Antinodal Lines . 44 3.5 Spectral Properties . 47 3.6 Neumann Function . 47 3.7 Related Structures . 48 3.7.1 Hemiequilateral Triangle . 48 3.7.2 Regular Rhombus . 49 3.7.3 Regular Hexagon . 50 3.8 Carl Gottfried Neumann . 50 4 Polygons with Trigonometric Eigenfunctions 52 4.1 Complete Set of Trigonometric Eigenfunctions . 53 4.2 Partial Set of Trigonometric Eigenfunctions . 57 4.3 Trigonometric Eigenfunctions under Mixed Boundary Conditions 60 4.4 Friedrich Carl Alwin Pockels . 61 5 Modal Degeneracy 63 5.1 Equilateral Triangular Modes . 64 5.2 Modal Degeneracy: Questions . 65 5.3 Eisenstein Primes . 66 5.4 Modal Degeneracy: Answers . 69 5.5 Modal Degeneracy: Examples . 70 5.6 Gotthold Eisenstein . 71 6 The Radiation Boundary Condition 73 6.1 The Robin Eigenproblem for the Equilateral Triangle . 74 6.2 Construction of Modes . 74 6.2.1 Symmetric Modes . 75 6.2.2 Antisymmetric Modes . 78 6.3 Modal Properties . 79 6.4 Spectral Properties . 82 6.5 Orthogonality . 84 6.6 Completeness . 85 6.7 Robin Function . 87 6.8 Victor Gustave Robin . 89 Preface ix 7 The Absorbing Boundary Condition 91 7.1 The Absorbing Eigenproblem for the Equilateral Triangle . 92 7.2 Symmetric/Antisymmetric Modes . 93 7.3 Modal Properties . 95 7.4 The Limit σ ......................... 97 → −∞ 7.4.1 ABC-Dirichlet Modes . 97 7.4.2 The Missing Modes . 98 7.5 Spectral Properties . 111 7.6 Orthogonality and Completeness . 112 7.7 Hilbert and Courant . 114 7.7.1 David Hilbert . 115 7.7.2 Richard Courant . 117 8 The Sturm-Liouville Boundary Value Problem 120 8.1 Solution of the Sturm-Liouville Boundary Value Problem . 121 8.2 S-L BVP Solution Properties . 123 8.3 The Case of Real σ . 124 8.3.1 The Case σ 0 . 125 ≥ 8.3.2 The Case σ < 0 . 125 8.4 The Case of Complex σ . 128 8.4.1 The Case Re(σ) 0 . 134 ≥ 8.4.2 The Case Re(σ) < 0 .