Phononic Crystals. By Vincent Laude. De Gruyter, 2015. Pp. XII+427. Price (hardcover) Euro 139.95, USD 196.00, GBP 104.99. ISBN 978‐3‐11‐030265‐3.

The monograph Phononic Crystals by Vincent Laude has appeared recently. It is very timely and describes the physical and mathematical basics for structures with periodic acoustic impedances. I start the review of this excellent monograph with comments concerning the term ‘phononic crystals’. The 21st century has been marked by the rapid development of nanotechnologies and nanoengineering. Various types of periodic structures have been designed, fabricated and used. These structures, artificially made by humans using advanced bottom-up or top-down fabrication methods of nanotechnologies, are often named by a quantum term depending on the physical nature of the quanta and the area of their applications. For elastic waves, in quantum terms phonons, the artificial structures with periodic distribution of the acoustic impedance are known as ‘phononic crystals’ in analogy with ‘photonic crystals’ for electromagnetic waves (i.e. photons). The essential feature of phononic crystals is the existence of a ‘phononic band gap’, which does not allow the propagation of elastic waves with the frequency falling into this gap. Actually, this remarkable property of artificial periodic structures for sound was known and used for many years before the term ‘phononic crystals’ was introduced at the end of the 20th century. Perhaps, for low-frequency sound, it would be more reasonable to call these structures ‘sonic’ or ‘acoustic’ crystals, but the name is unimportant when we are considering the physical background and equations describing the behavior of elastic waves in objects with periodic acoustic impedance. This unique book by Vincent Laude is the first monograph that is concerned exclusively with phononic crystals, and it summarizes the theoretical development in this wide and nowadays popular field. In the book the reader will find various theoretical, analytical and numerical approaches for describing the properties of phononic crystals. The author accompanies the text and mathematical equations by experimental figures, and corresponding references, which make the book extremely useful not only for researchers interested in the theory of phononic crystals but also for experimentalists. The book will be useful for scientists in solid state physics at all levels interested in topics ranging from basic one-dimensional elastic equations and simple analytical solutions up to complex algorithms of numerical calculations for twoand three-dimensional periodic structures. The book is also appropriate for postgraduate students who are working on their projects and dissertations and early career researchers who are just entering the topic in their experimental or theoretical activity. Advanced readers will find many useful sections describing various types of phononic crystals and related phenomena. The book consists of three parts occupying around 400 pages. The first part describes the case of scalar waves in periodic media. For clarity, the author uses in this case the terms ‘acoustic’ waves and ‘sonic’ crystals. The first sections give the theoretical basics for scalar waves and Bloch’s theorem and show the formation of band gaps. These well known topics, described in various textbooks on solid state physics, are combined together in a very compact and clear way, opening the gate to the field of phononic crystals for beginners. Using the example of one-dimensional periodic structures the author shows how to obtain analytical solutions, giving a clear feeling for the parameters that define the acoustic properties of an object. Turning to twoand three-dimensional cases, the author points to the complexity of the problem and the necessity of using numerical methods which become the main theoretical instruments in the next parts of ISSN 1600-5767