Mechanics of musical instruments
[The articles of this volume will not be indexed individually.] The theory of musical instruments has represented since almost three centuries one of the most fascinating fields where the methods of mechanics have proved their power in order to understand and explain the working of apparatuses that for many centuries were exclusively commited to the skill of families of artisans. At the beginning of the eighteenth century, the propagation of sound through vibrating strings, air tubes, membranes, plates, shells became a problem describable by rigorous equations, able to interpret and predict the generation of sounds and their variations due to the shape of each instrument. People like Daniel Bernoulli, d'Alembert, Euler, Chladni, gave fundamental contributions to a rational theory of sound, under the unifying concept of linear wave propagation in an elastic continuum. A collection of these results and a new variational formulation of the problem of wave propagation was proposed by Rayleigh in his famous treatise ``The theory of sound'' (1894). But the response of musical instruments is not perfectly conformal to the linear model. On one hand, nonlinearity is an essential aspect of many instruments, in the sense that their dynamical responses (playability) are heavily modified by the elastic deformations. On the other hand, dissipation mechanisms are necessary for understanding decay transients or radiation. The purpose of this volume is to offer the beginner a gradual introduction to musical acoustics. The matter is ordered in seven chapters, each written by a specialist, ranging from pure mathematical theory of the simple oscillator to problems which are specific to musical instruments. The first four chapters are a compendium of classic notions about mechanical vibrations of systems with a finite number of degrees of freedom, of strings, beams, and plates. Unfortunately, there are some overlapping and some omissions. Among them the chapter one, written by \textit{P. Hagedorn}, is the most systematic and rigorous. Chapter five treats the working of bowed string instruments like the violin. The sound of a violin is not due to simple vibrations of a set of elastic strings fixed at their ends, because the emitted sound can be controlled by the player through its duration. This chapter, however, is more a survey of recent results in the numerical simulation of the bowed-string model rather than a section of a collection of lecture notes. The same characteristic is possessed by the subsequent chapter, dedicated to reed instruments. Even under the simplifying assumption that damping and inertia are neglegible, the equations governing the production of sound of a cylindrical clarinet are highly nonlinear. The final chapter treats, more generally, aeroacoustic phenomena. Here the exposition is well organised and clear, as requested for an introductory book. The equations are the classical ones, and the consequences of the discussion of their solutions are surprisingly appealing. As a final comment, the subject is very interesting, but a more uniform arrangement of the matter would be auspicable in a future edition of the book.
mechanical vibrations of systems with finite number of degrees of freedom
scientific article; zbMATH DE number 904958