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The Approximate Solution of One-Dimensional Wave Equations

The Approximate Solution of One-Dimensional Wave Equations Wave Equations* CARL ECKART University of California, Marine Physical Laboratory, San Diego, California CONTENTS Introduction 1. Examples of Wave Equations ................................................. 400 2. The Hamiltonian Functions and the Logarithmic Decrements .. .e. 401 3. The Fourier Solution of the Initial Value Problem ............................... 402 4. The Method of Stationary Phase, and the Eikonal ..................... ........ 403 5. Group Velocity and the Hamilton-Jacobi Function .............................. 404 6. The Resolution of the Waves into a Spectrum .406 7. Wave Fronts .406 8. Wave Fronts of Constant Velocity, First Kind .408 9. Wave Fronts of Constant Velocity, Second Kind .409 10. Unresolved Waves . 410 11. The Boundary Value Problem ......................... 411 12. Formal Theory of Multiple Hamiltonians .412 13. Arbitrariness of the Method of Stationary Phase .414 14. The Method of Steepest Descent .415 15. The Remainder in the Method of Stationary Phase .415 INTRODUCTION THE rigorous and general solution of linear partial differential equations with constant coefficients can be obtained without difficulty, in the form of a Fourier integral. 2 However, the very generality of the Fourier integral makes it difficult to interpret in any detail, unless it can be evaluated in terms of elementary functions. The need for http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Reviews of Modern Physics American Physical Society (APS)

The Approximate Solution of One-Dimensional Wave Equations

Eckart, Carl
Reviews of Modern Physics , Volume 20 (2) – Apr 1, 1948
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Publisher
American Physical Society (APS)
Copyright
Copyright © 1948 The American Physical Society
ISSN
1539-0756
DOI
10.1103/RevModPhys.20.399
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Abstract

Wave Equations* CARL ECKART University of California, Marine Physical Laboratory, San Diego, California CONTENTS Introduction 1. Examples of Wave Equations ................................................. 400 2. The Hamiltonian Functions and the Logarithmic Decrements .. .e. 401 3. The Fourier Solution of the Initial Value Problem ............................... 402 4. The Method of Stationary Phase, and the Eikonal ..................... ........ 403 5. Group Velocity and the Hamilton-Jacobi Function .............................. 404 6. The Resolution of the Waves into a Spectrum .406 7. Wave Fronts .406 8. Wave Fronts of Constant Velocity, First Kind .408 9. Wave Fronts of Constant Velocity, Second Kind .409 10. Unresolved Waves . 410 11. The Boundary Value Problem ......................... 411 12. Formal Theory of Multiple Hamiltonians .412 13. Arbitrariness of the Method of Stationary Phase .414 14. The Method of Steepest Descent .415 15. The Remainder in the Method of Stationary Phase .415 INTRODUCTION THE rigorous and general solution of linear partial differential equations with constant coefficients can be obtained without difficulty, in the form of a Fourier integral. 2 However, the very generality of the Fourier integral makes it difficult to interpret in any detail, unless it can be evaluated in terms of elementary functions. The need for

Journal

Reviews of Modern Physics – American Physical Society (APS)

Published: Apr 1, 1948

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