Asymptotic Wave Theory

1st Edition - January 1, 1976
Author: Maurice Roseau
Language: English
eBook ISBN:
9 7 8 - 0 - 4 4 4 - 6 0 1 9 1 - 9

Asymptotic Wave Theory investigates the asymptotic behavior of wave representations and presents some typical results borrowed from hydrodynamics and elasticity theory. It… Read more

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Asymptotic Wave Theory investigates the asymptotic behavior of wave representations and presents some typical results borrowed from hydrodynamics and elasticity theory. It describes techniques such as Fourier-Laplace transforms, operational calculus, special functions, and asymptotic methods. It also discusses applications to the wave equation, the elements of scattering matrix theory, problems related to the wave equation, and diffraction. Organized into eight chapters, this volume begins with an overview of the Fourier-Laplace integral, the Mellin transform, and special functions such as the gamma function and the Bessel functions. It then considers wave propagation, with emphasis on representations of plane, cylindrical or spherical waves. It methodically introduces the reader to the reflexion and refraction of a plane wave at the interface between two homogeneous media, the asymptotic expansion of Hankel's functions in the neighborhood of the point at infinity, and the asymptotic behavior of the Laplace transform. The book also examines the method of steepest descent, the asymptotic representation of Hankel's function of large order, and the scattering matrix theory. The remaining chapters focus on problems of flow in open channels, the propagation of elastic waves within a layered spherical body, and some problems in water wave theory. This book is a valuable resource for mechanics and students of applied mathematics and mechanics.

1. The Fourier-Laplace Integral 1.1. The Laplace transform 1.1.1. The direct problem 1.1.2. The inverse problem 1.1.3. Elementary rules 1.2. The Fourier transform in L1 1.3. The Fourier transform in L2 1.4. The Laplace transform (continued) 1.5. The Mellin transform2. Special Functions 2.1. The gamma function 2.1.1. A summation formula 2.1.2. The Eulerian definition of the function Γ(z) 2.1.3. The Laplace transform of tv 2.1.4. The relation between the function Γ(z) and the group of linear mappingsof the real line into itself 2.1.5. The error function 2.II. The Bessel functions 2.11.1. Definitions 2.11.2. The Kepler equation and Bessel functions 2.11.3. The group of displacements in the plane and Bessel's functions 2.11.4. The Bessel functions of purely imaginary argument 2.11.5. The Hankel functions 2.11.6. Addition formulae for Bessel's functions3. The Wave Equation 3.1. Introduction 3.2. The reflexion and refraction of a plane wave at the interface between twohomogeneous media 3.3. Spherical waves 3.4. Cylindrical waves 3.5. Group velocity 3.6. Wave guide 3.7. Successive reflexions of plane waves at two parallel rigid planes 3.8. The relation between spherical and plane waves 3.9. The reflexion of a spherical wave at a plane interface 3.10. An alternative approach to Weyl's formula. Poritsky's generalisation4. Asymptotic Methods 4.1. Asymptotic expansion 4.2. The asymptotic expansion of Hankel's functions in the neighborhood of thepoint at infinity 4.3. The Laplace method 4.4. Asymptotic relations and Laplace's transform 4.5. The Laplace method (continued) 4.6. The method of steepest descent 4.7. Waves in linear dispersive media 4.8. The asymptotic representation of the reflected wave in the problem of a sphericalwave impinging on a plane interface. The lateral wave 4.9. The method of steepest descent; an extension to the case when some pole islocated near the saddle 4.10. The asymptotic representation of Hankel's functions of large order 4.11. An asymptotic representation of Legendre's functions of large order 4.12. The asymptotic representation of Hankel's functions of large order (continued) 4.12.1. A discussion of the equation y+2ry=0 4.12.2. Approximate representations of Hv(1)(v), dHv(2)(z)/dz