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Vibration and damping in distributed systems. Vol. 2: WKB and wave methods, visualization and experimentation. (English) Zbl 0852.73002

Studies in Advanced Mathematics. Boca Raton, FL: CRC Press. xxvi, 367 p. (1993).
This is the second volume that the authors have dedicated to the theory of systems with distributed parameters, that is, more precisely, to the eigenvalues of linear partial differential equations relative to some systems of orthogonal curvilinear coordinates. If the equation is separable and the domain is a rectangle, then the eigenvalues and the eigenfunctions of the equation can be calculated by the method of separation of variables. However, it often happens that, still the domain being a rectangle, the separation of variables is not possible. Thus, starting from the first half of the 19th century, arose the need of finding asymptotic estimates of the eigenvalues when the coefficients, though variable, vary slowly. The problem was initially suggested by celestial mechanics, but later it turned out to be important in the motion of tides, in quantum mechanics, and in the theory of mechanical vibrations.
The authors divide the methods, which, in their opinion, are the most effective for evaluating asymptotically the eigenvalues, into three classes, corresponding to the first three chapters of the volume. In the first chapter they present methods valid for ordinary or partial differential equations reducible to ordinary ones by separation of variables. These methods are customarily called “approximation WBK” (Wentzel, Kramer, Brillouin), although their central idea was already proposed by Birkhoff and Langer in 1923. However, when one wants to evaluate the eigenvalues of partial differential equations on non-rectangular domains, the WBK method does not apply, and then one may take recourse to a method, introduced by Keller and Rubinow, based on a suitable representation of solutions in terms of phase functions of the operator.
There are, in addition, other methods for calculating eigenvalues: the classical method of variational type, of Rayleigh and Courant; the asymptotic method of Bolotin; other totally numerical methods. Some results obtained by the last method for plane domains have been collected in chapter 4. Finally, chapter 5, written by D. L. Russel, is dedicated to the theoretical and experimental analysis of the vibration of a beam immersed in a liquid and of a beam attached at one end with another beam of infinite length.
The book contains various interesting problems, illustrated by many examples. The application of the WBK method and of that by Keller and Rubinow are very detailed, but this is useful in order to understand how they work. Unfortunately, the organization of the matter is incoherent. For instance, the last three chapters which contain advanced methods, open to generalizations, are completely extraneus to the first two. The other chapters present well-known theories or even numerical solutions for cases not substantially different from each other. Finally, the chapter written by Russel should have been published separately.
Reviewer: P.Villaggio (Pisa)

MSC:

74-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to mechanics of deformable solids
74H45 Vibrations in dynamical problems in solid mechanics
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