Propagation of Waves and the Equations of Hydrodynamics (Progress in Mathematical Physics) 〈Vol. 57〉

Hadamard, J.

Birkhäuser（2009/10発売）

• 製本 Hardcover:ハードカバー版／ページ数 300 p.
• 商品コード 9783764399689

基本説明

First English translation of a real classic. Translator's notes review and summarize the impact of the original work and provide historical background information on the author.

Full Description

This seminal work of Hadamard on the mathematical theory of waves was written over 100 years ago and it continues to be cited as a reference by researchers in mathematical physics. The reason for the enduring interest in this book can be found in its legacy. The conception of waves as discontinuities in some level of derivative of a wave function that propagate along the bicharacteristics of the wave equation spawned many of the important advances to both the purely mathematical theory of hyperbolic equations, as well as the more physical and engineering-oriented treatments of the subject of wave motion. In mathematics, one can follow the implications of this work into the subsequent lectures that Hadamard gave on the Cauchy problem for linear partial differential equations. But one should regard this masterful treatise not only as a precursor to the later lectures on the Cauchy problem, but as a complementary work in which he establishes the roots of the mathematical theory in continuum mechanics.

Contents

Translator's notes. Forword.- 1. The second boundary-value problem of the theory of harmonic functions.- 1.1. Classical properties of harmonic functions.- 1.2. The second boundary problem. Existence of the solution.- 1.3. Case of the plane.- 1.4. Case of space. Application of Neumann's method.- 1.5. The functions of Fr. Neumann and Klein.- 1.6. Case of the sphere. 1.7. Mixed problems. 2. Waves from the kinematical viewpoint.- 2.1Classical results.- 2.2. Study of discontinuities. Identity conditions.- 2.3. Study of discontinuities (cont.). Kinematic compatibility conditions.- 2.4. Study of discontinuities (cont.). Higher-order compatibility conditions.- 3. The formulation of the hydrodynamical problem as an equation.- 3.1. The interior equations and a supplementary condition.- 3.2. Introduction of the boundary conditions.- 4. Rectilinear motion of gases.- 4.1. Case of constant propagation velocity.- 4.2. General case.- 4.3. The Riemann-Hugoniot phenomenon.- 5. Motions in space.- 6. Applications to the theory of elasticity.- 7. The general theory of characteristics.- 7.1. Characteristics and bicharacteristics.- 7.2. Existence theorems.- 7.3. The case of linear equations.- Note I. On the Cauchy problem and characteristics.- Note II. On slips in fluids.- Note III. On the vortices produced by shock waves.- Note IV. On reflection in the case of a fixed piston.