zbMATH — the first resource for mathematics

Vibration and coupling of continuous systems. Asymptotic methods. (English) Zbl 0698.70003
Berlin etc.: Springer-Verlag. xv, 421 p. DM 138.00 (1989).
Subject of this book are: linear vibration problems in connection with continuous systems, more complicated coupling problems by systems consisting of such subsystems, further oscillatory systems which can be considered as perturbation of a known simple system.
The 1st and 2nd chapters deal with the vibration theory of systems with discrete spectra and some important examples. The 3rd chapter contains a body of knowledge later applied in connection with operators and spectral theories. Equations of thermoelasticity are the subject of the 4th chapter, further viscoelastic systems described by integro-differential terms; these go partly beyond the previously outlined material, and partly can be led back to it by certain transformations. A physically easily applicable rigorous spectral perturbation theory is given in the 5th chapter. The 6th chapter contains heuristical asymptotic methods, the matched asymptotic expansion and the method of two scales. Subject of the 7th chapter are systems containing a small parameter, treated by perturbation theory, systems consisting of soft and stiff parts, effects of a small viscosity, systems with concentrated masses. The 8th and 9th chapters deal with acoustic vibrations in unbounded regions and their coupling with an elastic body. Spectral body of the Helmholtz equation in outer regions can be found in the 8th chapter, which treats the asymptotic behaviour of the solutions of the wave equations too for \(t\to \infty\) with special attention to the scattering frequencies. These results are applied in the 9th chapter for coupled systems, vibrations of an elastic body immersed in compressible or incompressible fluid, small- and large frequency phenomena and the Helmholtz-resonator. Numerous results appear in this book for the first time in printed form. A number of open questions and suggested research themes are enumerated.
It is felt: applied mathematicians, physicists and engineers “waited” for this book some years ago, and now it appeared finally causing a great pleasure. The book will take a very important place in the libraries of such persons and institutes.
Reviewer: Á.Bosznay

70-02 Research exposition (monographs, survey articles) pertaining to mechanics of particles and systems
70J99 Linear vibration theory
74-02 Research exposition (monographs, survey articles) pertaining to mechanics of deformable solids
74H45 Vibrations in dynamical problems in solid mechanics
74A15 Thermodynamics in solid mechanics
74A99 Generalities, axiomatics, foundations of continuum mechanics of solids
76D99 Incompressible viscous fluids