Linear and nonlinear waves

The study of waves can be traced back to antiquity where philosophers, such as Pythagoras (c.560-480 BC), studied the relation of pitch and length of string in musical instruments. However, it was not until the work of Giovani Benedetti (1530-90), Isaac Beeckman (1588-1637) and Galileo (1564-1642) that the relationship between pitch and frequency was discovered. This started the science of acoustics, a term coined by Joseph Sauveur (1653-1716) who showed that strings can vibrate simultaneously at a fundamental frequency and at integral multiples that he called harmonics. Isaac Newton (1642-1727) was the first to calculate the speed of sound in his Principia. However, he assumed isothermal conditions so his value was too low compared with measured values. This discrepancy was resolved by Laplace (1749-1827) when he included adiabatic heating and cooling effects. The first analytical solution for a vibrating string was given by Brook Taylor (1685-1731). After this, advances were made by Daniel Bernoulli (1700-82), Leonard Euler (1707-83) and Jean d’Alembert (1717-83) who found the first solution to the linear wave equation, see section (3.2). Whilst others had shown that a wave can be represented as a sum of simple harmonic oscillations, it was Joseph Fourier (1768-1830) who conjectured that arbitrary functions can be represented by the superposition of an infinite sum of sines and cosines now known as the Fourier series. However, whilst his conjecture was controversial and not widely accepted at the time, Dirichlet subsequently provided a proof, in 1828, that all functions satisfying Dirichlet’s conditions (i.e. non-pathological piecewise continuous) could be represented by a convergent Fourier series. Finally, the subject of classical acoustics was laid down and presented as a coherent whole by John William Strutt (Lord Rayleigh, 1832-1901) in his treatise Theory of Sound. The science of modern acoustics has now moved into such diverse areas as sonar, auditoria, electronic amplifiers, etc.

[1]  T. N. Stevenson,et al.  Fluid Mechanics , 2021, Nature.

[2]  B. V. Leer,et al.  Towards the ultimate conservative difference scheme V. A second-order sequel to Godunov's method , 1979 .

[3]  J. Hadamard,et al.  Lectures on Cauchy's Problem in Linear Partial Differential Equations , 1924 .

[4]  Peter Walker.  Chambers dictionary of science and technology , 2000 .

[5]  L. Sedov.  Similarity and Dimensional Methods in Mechanics , 1960 .

[6]  D. Korteweg,et al.  On the change of form of long waves advancing in a rectangular canal, and on a new type of long stationary waves , 2011 .

[7]  A. V. Manzhirov,et al.  Handbook of mathematics for engineers and scientists , 2006 .

[8]  William E. Schiesser,et al.  Traveling Wave Analysis of Partial Differential Equations: Numerical and Analytical Methods with Matlab and Maple , 2011 .

[9]  H. J.,et al.  Hydrodynamics , 1924, Nature.

[10]  Victor A. Galaktionov,et al.  Exact Solutions and Invariant Subspaces of Nonlinear Partial Differential Equations in Mechanics and Physics , 2006 .

[11]  J. Borrero,et al.  Field Data and Satellite Imagery of Tsunami Effects in Banda Aceh , 2005, Science.

[12]  C. Hirsch.  Numerical Computation of Internal and External Flows, Volume 2: Computational Methods for Inviscid and Viscous Flows , 1990 .

[13]  H. H. Dai,et al.  Periodic wave solutions of nonlinear equations by Hirota's bilinear method , 2006 .

[14]  P. Sachdev,et al.  The Blast Wave , 2004 .

[15]  William F. Bynum,et al.  Dictionary of the History of Science , 2014 .

[16]  W. Rindler,et al.  Gravitation and Spacetime , 1977 .

[17]  Geoffrey Ingram Taylor,et al.  The formation of a blast wave by a very intense explosion. - II. The atomic explosion of 1945 , 1950, Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences.

[18]  Chi-Wang Shu.  Essentially non-oscillatory and weighted essentially non-oscillatory schemes for hyperbolic conservation laws , 1998 .

[19]  Sergei A. Tretyakov,et al.  Millimeter-Wave Waveguides , 2003 .

[20]  F. Kneubühl,et al.  Oscillations and Waves , 1985 .

[21]  J. Z. Zhu,et al.  The finite element method , 1977 .

[22]  R. Batchelor,et al.  Chemical waves , 1984, Nature.

[23]  R. LeVeque.  Finite Volume Methods for Hyperbolic Problems: Characteristics and Riemann Problems for Linear Hyperbolic Equations , 2002 .

[24]  W. C. Elmore,et al.  Physics of Waves , 1969 .

[25]  K. Okamoto.  Fundamentals of Optical Waveguides , 2000 .

[26]  J. M. Lackie.  Chambers dictionary of science and technology , 2007 .

[27]  T. I. Valchev,et al.  How many types of soliton solutions do we know , 2007, 0708.1253.

[28]  L. C. Wrobel.  Numerical computation of internal and external flows. Volume 2: Computational methods for inviscid and viscous flows , 1992 .

[29]  P. Wesseling.  Principles of Computational Fluid Dynamics , 2000 .

[30]  D. Peregrine.  A Modern Introduction to the Mathematical Theory of Water Waves. By R. S. Johnson. Cambridge University Press, 1997. xiv+445 pp. Hardback ISBN 0 521 59172 4 £55.00; paperback 0 521 59832 X £19.95. , 1998, Journal of Fluid Mechanics.

[31]  E. M. de Jager.  On the origin of the Korteweg-de Vries equation , 2011 .

[32]  G. Rowlands,et al.  Nonlinear Waves, Solitons and Chaos , 1990 .

[33]  Anne H. Soukhanov.  Encarta World English Dictionary , 1999 .

[34]  L. Chambers.  Linear and Nonlinear Waves , 2000, The Mathematical Gazette.

[35]  R. Courant,et al.  Methods of Mathematical Physics , 1962 .

[36]  N. M. Queen,et al.  Oscillations and Waves , 1991 .

[37]  I. Bohachevsky,et al.  Finite difference method for numerical computation of discontinuous solutions of the equations of fluid dynamics , 1959 .

[38]  Andrei D. Polyanin,et al.  Partial differential equation , 2008, Scholarpedia.

[39]  A. Polyanin,et al.  Handbook of First-Order Partial Differential Equations , 2001 .

[40]  W. Schiesser.  Computational Mathematics in Engineering and Applied Science: ODEs, DAEs, and PDEs , 1993 .

[41]  E. Toro.  Shock-Capturing Methods for Free-Surface Shallow Flows , 2001 .

[42]  Samir Hamdi,et al.  Method of lines , 2007, Scholarpedia.

[43]  C. Hirsch,et al.  Numerical Computation of Internal and External Flows. By C. HIRSCH. Wiley. Vol. 1, Fundamentals of Numerical Discretization. 1988. 515 pp. £60. Vol. 2, Computational Methods for Inviscid and Viscous Flows. 1990, 691 pp. £65. , 1991, Journal of Fluid Mechanics.

[44]  J. Miller.  Numerical Analysis , 1966, Nature.

[45]  A. Choudary,et al.  Partial Differential Equations An Introduction , 2010, 1004.2134.

[46]  P. Drazin.  SOLITONS, NONLINEAR EVOLUTION EQUATIONS AND INVERSE SCATTERING (London Mathematical Society Lecture Note Series 149) , 1993 .

[47]  H. Kreiss,et al.  Initial-Boundary Value Problems and the Navier-Stokes Equations , 2004 .

[48]  Ji-Huan He,et al.  Exp-function method for nonlinear wave equations , 2006 .

[49]  G. Taylor.  The formation of a blast wave by a very intense explosion I. Theoretical discussion , 1950, Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences.

[50]  P. Drazin,et al.  Solitons: An Introduction , 1989 .

[51]  Adam Ostaszewski,et al.  Advanced Mathematical Methods , 1990 .

[52]  E. Toro.  Riemann Solvers and Numerical Methods for Fluid Dynamics , 1997 .

[53]  T. Tao.  Nonlinear dispersive equations : local and global analysis , 2006 .

[54]  Ronald Fedkiw,et al.  Level set methods and dynamic implicit surfaces , 2002, Applied mathematical sciences.

[55]  Chauncey D. Leake,et al.  British Association for the Advancement of Science , 1953, Science.

[56]  W. Malfliet.  Solitary wave solutions of nonlinear wave equations , 1992 .

[57]  R. Pletcher,et al.  Computational Fluid Mechanics and Heat Transfer. By D. A ANDERSON, J. C. TANNEHILL and R. H. PLETCHER. Hemisphere, 1984. 599 pp. $39.95. , 1986, Journal of Fluid Mechanics.

[58]  A. Polyanin,et al.  Handbook of Nonlinear Partial Differential Equations , 2003 .

[59]  R. LeVeque,et al.  FINITE VOLUME METHODS AND ADAPTIVE REFINEMENT FOR GLOBAL TSUNAMI PROPAGATION AND LOCAL INUNDATION. , 2006 .

[60]  Athanassios S. Fokas,et al.  Hodograph transformations of linearizable partial differential equations , 1989 .

[61]  James A. Sethian,et al.  Level Set Methods and Fast Marching Methods , 1999 .

[62]  F. Cajori.  A history of mathematics , 1989 .

[63]  Willy Hereman,et al.  The Tanh Method : A Tool to Solve Nonlinear Partial Differential Equations with Symbolic Software , 2005 .

[64]  W. Hereman,et al.  The tanh method: I. Exact solutions of nonlinear evolution and wave equations , 1996 .

[65]  Marko Subasic,et al.  Level Set Methods and Fast Marching Methods , 2003 .

[66]  A. E. Gill.  Atmosphere-Ocean Dynamics , 1982 .

[67]  M. Ablowitz,et al.  Solitons, Nonlinear Evolution Equations and Inverse Scattering , 1992 .

[68]  David J. Kaup,et al.  How Many Types of Soliton Solutions do We Know , 2006 .

[69]  R. Courant,et al.  Methods of Mathematical Physics, Vol. I , 1954 .

[70]  Toby Gelfand.  Dictionary of the History of Science , 1982 .

[71]  V. Arnold.  Mathematical Methods of Classical Mechanics , 1974 .

[72]  D. Korteweg,et al.  XLI. On the change of form of long waves advancing in a rectangular canal, and on a new type of long stationary waves , 1895 .

[73]  R. S. Amando.  Numerical computation of internal and external flows, volume 1: Fundamentals of numerical discretization , 1989 .

[74]  Chi-Wang Shu,et al.  High Order Weighted Essentially Nonoscillatory Schemes for Convection Dominated Problems , 2009, SIAM Rev..

[75]  Terence Tao,et al.  Why are solitons stable , 2008, 0802.2408.

[76]  K J Whiteman,et al.  Linear and Nonlinear Waves , 1975 .

[77]  James D. Murray.  Mathematical Biology: I. An Introduction , 2007 .

[78]  O. Cornejo-Perez,et al.  Nonlinear Second Order Ode's: Factorizations and Particular Solutions , 2005 .

[79]  Mohamed A. Helal,et al.  Solitons, Introduction to , 2009, Encyclopedia of Complexity and Systems Science.

[80]  A. Shadowitz,et al.  The electromagnetic field , 1974 .

[81]  Roger Knobel,et al.  An introduction to the mathematical theory of waves , 1999 .

[82]  Ozer Igra,et al.  Handbook of shock waves , 2001 .

[83]  R. A. Cairns.  Nonlinear Waves, Solitons and Chaos, 2nd edition, by Eryk Infeld and George Rowlands. Cambridge University Press, 2000. Hardback: ISBN 0 521 63212 9. £75.00, $120.00. Paperback: ISBN 0 521 63557 8. £27.95, $44.95. , 2001 .

[84]  William E. Schiesser,et al.  A Compendium of Partial Differential Equation Models: Method of Lines Analysis with Matlab , 2009 .

[85]  O. Zienkiewicz.  The Finite Element Method In Engineering Science , 1971 .

[86]  A. Aksenov,et al.  CRC Handbook of Lie Group Analysis of Differential Equations. Vol. 2. Applications in Engineering and Physical Sciences , 1995 .