This book contains a selection of advanced topics suitable for final year undergraduates in science and engineering, and is based on courses of lectures given by one of us (G. S.) to various groups of third year engineering and science students at Imperial College over the past 15 years. It is assumed that the student has a good understanding of basic ancillary mathematics. The emphasis in the text is principally on the analytical understanding of the topics which is a vital prerequisite to any subsequent numerical and computational work. In no sense does the book pretend to be a comprehensive or highly rigorous account, but rather attempts to provide an accessible working knowledge of some of the current important analytical tools required in modern physics and engineering. The text may also provide a useful revision and reference guide for postgraduates.

Each chapter concludes with a selection of problems to be worked, some of which have been taken from Imperial College examination papers over the last ten years. Answers are given at the end of the book.

We wish to thank Dr Tony Dowson and Dr Noel Baker for reading the manuscript and making a number of helpful suggestions.

The simple resonators that have been the principal theme up to this point possessed one resonant mode only and were, moreover, excited by external influences whose reaction to the effect they produced could be ignored. We turn now to extensions of these ideas in two directions. First, the resonator may consist of a string, a tube, a transmission line, a waveguide – all media for the propagation of waves, whether mechanical, acoustic or electromagnetic – and may therefore be capable of excitation in a number of different modes (see the discussion of the vibrations of a string in chapter 2). Secondly, the resonator, which may be one of the above or something simpler, may be embedded in a medium for wave-propagation, being excited by the incidence of a wave and re-radiating a wave as it responds to the excitation. For the most part we shall confine the discussion to onedimensional systems, with only occasional excursions into the considerably more versatile and complicated realm of three dimensions, such as the response of a small resonant system to a plane wave.

Preliminary remarks about one-dimensional waves; characteristic impedance and admittance

A well-developed calculus exists for treating the reflection and transmission of waves on one-dimensional transmission lines, and this is unquestionably the right approach to adopt in engineering design and in the analysis of all but the most elementary problems.

It has long been known that a resonant system can be set into oscillation by periodically varying the parameters. A very extensive analysis was given by Rayleigh, who drew attention to analogies with rather different physical processes – the perturbation of a planetary orbit by another planet having a quite different period in some near-integral relationship, and the propagation of waves in periodically stratified media. This latter example relates to a phenomenon of much greater physical interest now than in Rayleigh's day since it is basic to Bragg reflection of X-rays and to the motion of electrons in solids. Nevertheless, these are analogies only in the sense that the same type of differential equation is involved in all of them, as well as in the theory of vibrating elliptical membranes. It can hardly be claimed that the solution of one of these problems adds anything to the intuitive understanding of another. A mathematical framework, however, is well worth having, for these are not easy problems. The examples presented here earn a place in their own right as showing yet another aspect of the variety of oscillatory phenomena in nature. At the same time, there are important technological applications in the form of parametric amplifiers, as will be discussed in outline at the end of the chapter.

Probably the easiest way to demonstrate parametric excitation is to set up a simple pendulum whose length can be slightly varied at twice the natural frequency.

This chapter is concerned only with simple systems in steady oscillation, such as were classified in chapter 2 under the heading of negative resistance devices, feedback oscillators and relaxation oscillators. Here we shall attempt to refine the description and classification of the different types though, as is common in such attempts, firm divisions are hard to find. At the same time we shall analyse a number of examples so as to understand what conditions must be satisfied for them to oscillate spontaneously, how the amplitude of oscillation is limited by non-linearity, and what determines the ultimate waveform. No attempt will be made to establish rigorously the general conditions for oscillation to occur. This is an important and well-studied problem, but one which deserves the fuller treatment that will be found in specialized texts.

It has already been remarked, in chapter 2 and elsewhere, that a resonant system governed by a second-order equation such as (2.23) will oscillate spontaneously if k is negative. A source of energy is required to overcome inevitable dissipative effects, and among the many examples of how the energy may be injected probably the commonest is by feedback. Let us start, then, with a general survey of the feedback principle, with particular reference to the influence feedback may have on the performance of a resonant system, and not solely in setting it in spontaneous oscillation. The argument will be conducted in terms of electrical circuits, which account for the overwhelming majority of applications.

The plan and purpose of this book are outlined in chapter 1, and what is said there need not be repeated here. A preface does, however, offer the opportunity of acknowledging the help I have received during its preparation, and indeed over the years before it was even conceived. I cannot begin to guess how far my understanding and opinions have benefited from the good fortune that has allowed me to spend so much of my working life in the Cavendish Laboratory, surrounded by physics and physicists of the highest quality, and by students who at their best were at least the equal of their teachers. For the resulting gifts of learning, casually offered and accepted for the most part, and now untraceable to their source, I extend belated thanks. More specifically, in the preparation of the material I have relied heavily on Douglas Stewart for the construction of apparatus, Christopher Nex for programming the computer to draw many of the diagrams (and incidentally for two notes on his bassoon in fig. 5.8), Gilbert Yates for help in the theory and practice of electronics, and Shirley Fieldhouse for typing and retyping of the manuscript. My thanks to each of them for much willing and expert help, and to the following who by discussion and criticism, by assistance in carrying out experiments, or by providing material for diagrams have given support and comfort on the way: Dr J. E. Baldwin, Mr W. E. Bircham, Prof. J. Clarke, Dr M. H. Gilbert, Prof. O. S. Heavens, Dr R. E. Hills, Dr I. P. Jones, Mr P. Jones, Mr P. Martel, Prof. D. H. Martin, Mr P. D. Maskell, Dr P. J. Mole, Mr E. Puplett, Dr G. P. Wallis and Mr D. K. Waterson.

The simple harmonic oscillator, driven by a sinusoidally varying force, is central to the discussion of vibrating systems, being a model for so many real systems and therefore serving to unify the description of very diverse physical problems. In view of this it is worth spending some time examining it from several different aspects, even though it might be thought that a formal solution of the equation of motion contained everything useful to be said on the matter. Indeed, if one were concerned only with physical systems that could be modelled exactly in these terms a single comprehensive treatment would suffice for all. Real systems, however, normally only approximate to this idealization, and alternative approaches may then prove their worth in allowing the behaviour to be apprehended semiintuitively, often enough with sufficient exactitude to make mathematical analysis unnecessary. The reader who has progressed to this point will be familiar enough with the most elementary analyses not to be worried that we approach the problem indirectly, picking up an argument that has already been partially developed. More familiar treatments will be introduced in due course.

Transfer function, compliance, susceptibility, admittance, impedance

The essential framework for this approach has already been laid down in chapter 5, where the concept of the transfer function χ(ω) was introduced. We had in mind there a linear transducer into which a sinusoidal signal A e−iωt was fed, and from which emerged an output signal χ(ω) A e−iωt.