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Stochastic wave propagation
Vol. 25 (1988)
REPORTS
ON
MATHEMATICAL
BOOK
PHYSICS
No. 3
REVIEWS
KAZIMIERZ SOBCZYK: Stochastic Wave Propagation, Polish Warszawa, Elsevier, Amsterdam-Oxford-New York-Tokyo,
Scientific Publishers, 1984, VIII + 248 pp.
Kazimierz Sobczyk (born 1939) is a well-known applied mathematician working in the Institute of Fundamental Technological Research of the Polish Academy of Sciences in Warsaw. After graduation in mathematics from the Warsaw University (1960) he started to work at this Institute and he went there through steps of doctor, docent and professor. But apart from this long professional career in one institution, he obtained a broad scientific experience in many foreign centers of applied mathematics (Courant Institute of Mathematical Sciences at New York University, MIT, the Glasgow University, Technical University of Denmark, etc.). Now we see the result of his long and consequent research, devoted mainly to the stochastic methods in mechanics, in form of a monograph on stochastic wave propagation. The book consists of 5 chapters, of which the first three present a more or less well-known material, and the last two show among contributions of other authors mainly results of his own research. The chapters are as follows: 1) mathematical preliminaries, especially, on stochastic differential equations and functional derivatives, 2) stochastic media: models and analysis, especially, hierarchy equations for moments and the Bethe-Salpeter equation, 3) wave propagation in continuous stochastic media, especially, diffusion approximation, functional approach and diagram technique, 4) wave propagation in discrete stochastic media, especially, layered media, scattering on randomly distributed inclusions, reverberation processes, 5) scattering of waves at stochastic surfaces, especially, Rayleigh and Kirchhoff methods, surfaces with random impedence. At the end a detailed bibliography containing 300 positions is given, among them of his own (he published over 40 original papers on random vibrations and stochastic wave propagation and 3 books). The book gives a very clear and elementary presentation of the actual state of the art in this field. The scope of the book is very broad: he considers elastic, thermoelastic, electromagnetic and magnetoelastic waves in very different kinds of media of various structure. So stochasticity comes from many different sources and behaves in various ways. The methods used for treating these problems are very modern and the treatment is as rigorous as possible in different applications. There is no pure mathematics for its own sake, no strictly mathematical style of postulates, [389]
390
BOOK
REVIEWS
definitions, theorems and proofs. Nevertheless, the presentation is mathematically correct and clear cut. The approximations used are systematically and consecutively presented, e.g. the Born approximation, the approximation of geometrical optics, the Rytov approximation, the parabolic approximation, the smoothing method, etc. The book may be used by a wide range of readers. Except for applied mathematicians, the book is addressed in the first line to people working in acoustics, radiophysics, optics, geophysics, mechanics. It presents a lot of new results of the author, and a modernized presentation of older results. As regards mathematical physicists, also these readers may find many interesting topics for themselves. The most important perhaps for this category of readers is the approach of stochastic equations, and such more abstract methods as the functional method, the Feynman-Kac formula, the Wiener integral or the Feynman graphs. The author uses freely many physical results, such as the Bethe-Salpeter equation or the Dyson equation, and many older classical methods, as those of Kirchhof and Rayleigh. Physicists may look in vain for some other more subtle methods, e.g. as those of the boundary wave of Rubinowicz, etc., but it is hard to have all in a monograph of this size. A deeper requirement, perhaps, would be to use also some more intrinsic physical concepts, as those of energy, entropy, temperature (of waves, not of the media!). Unfortunately, we rather do not see them. That is perhaps typical for authors coming from the side of mathematics. If they use statistics, probability theory, measure theory, stochastic processes, they seem to be on a safe ground, and no thermodynamics or even information theory is usually felt as lacking. But the reviewer hopes that, maybe, when the second edition of this excellent book will be prepared in future, the author will extend his interests and presentation also to these border lands between physics and mathematics. R. S. Ingarden
REPORTS
ON
MATHEMATICAL
BOOK
PHYSICS
No. 3
REVIEWS
KAZIMIERZ SOBCZYK: Stochastic Wave Propagation, Polish Warszawa, Elsevier, Amsterdam-Oxford-New York-Tokyo,
Scientific Publishers, 1984, VIII + 248 pp.
Kazimierz Sobczyk (born 1939) is a well-known applied mathematician working in the Institute of Fundamental Technological Research of the Polish Academy of Sciences in Warsaw. After graduation in mathematics from the Warsaw University (1960) he started to work at this Institute and he went there through steps of doctor, docent and professor. But apart from this long professional career in one institution, he obtained a broad scientific experience in many foreign centers of applied mathematics (Courant Institute of Mathematical Sciences at New York University, MIT, the Glasgow University, Technical University of Denmark, etc.). Now we see the result of his long and consequent research, devoted mainly to the stochastic methods in mechanics, in form of a monograph on stochastic wave propagation. The book consists of 5 chapters, of which the first three present a more or less well-known material, and the last two show among contributions of other authors mainly results of his own research. The chapters are as follows: 1) mathematical preliminaries, especially, on stochastic differential equations and functional derivatives, 2) stochastic media: models and analysis, especially, hierarchy equations for moments and the Bethe-Salpeter equation, 3) wave propagation in continuous stochastic media, especially, diffusion approximation, functional approach and diagram technique, 4) wave propagation in discrete stochastic media, especially, layered media, scattering on randomly distributed inclusions, reverberation processes, 5) scattering of waves at stochastic surfaces, especially, Rayleigh and Kirchhoff methods, surfaces with random impedence. At the end a detailed bibliography containing 300 positions is given, among them of his own (he published over 40 original papers on random vibrations and stochastic wave propagation and 3 books). The book gives a very clear and elementary presentation of the actual state of the art in this field. The scope of the book is very broad: he considers elastic, thermoelastic, electromagnetic and magnetoelastic waves in very different kinds of media of various structure. So stochasticity comes from many different sources and behaves in various ways. The methods used for treating these problems are very modern and the treatment is as rigorous as possible in different applications. There is no pure mathematics for its own sake, no strictly mathematical style of postulates, [389]
390
BOOK
REVIEWS
definitions, theorems and proofs. Nevertheless, the presentation is mathematically correct and clear cut. The approximations used are systematically and consecutively presented, e.g. the Born approximation, the approximation of geometrical optics, the Rytov approximation, the parabolic approximation, the smoothing method, etc. The book may be used by a wide range of readers. Except for applied mathematicians, the book is addressed in the first line to people working in acoustics, radiophysics, optics, geophysics, mechanics. It presents a lot of new results of the author, and a modernized presentation of older results. As regards mathematical physicists, also these readers may find many interesting topics for themselves. The most important perhaps for this category of readers is the approach of stochastic equations, and such more abstract methods as the functional method, the Feynman-Kac formula, the Wiener integral or the Feynman graphs. The author uses freely many physical results, such as the Bethe-Salpeter equation or the Dyson equation, and many older classical methods, as those of Kirchhof and Rayleigh. Physicists may look in vain for some other more subtle methods, e.g. as those of the boundary wave of Rubinowicz, etc., but it is hard to have all in a monograph of this size. A deeper requirement, perhaps, would be to use also some more intrinsic physical concepts, as those of energy, entropy, temperature (of waves, not of the media!). Unfortunately, we rather do not see them. That is perhaps typical for authors coming from the side of mathematics. If they use statistics, probability theory, measure theory, stochastic processes, they seem to be on a safe ground, and no thermodynamics or even information theory is usually felt as lacking. But the reviewer hopes that, maybe, when the second edition of this excellent book will be prepared in future, the author will extend his interests and presentation also to these border lands between physics and mathematics. R. S. Ingarden