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.

The natural line broadening resulting from electromagnetic or acoustic radiative processes, the only causes of broadening discussed so far, by no means exhausts the mechanisms available and indeed is usually so minor an effect as to be of small practical importance. We may distinguish broadening due to different behaviour on the part of different members of an ensemble from broadening exhibited by each member on its own. In the first class are Doppler broadening and broadening due to variations in environment, to which may be added the effects of slight differences between superficially similar systems (e.g. different isotopes). In the second class, in addition to radiative processes, must be counted anything, especially collision with other atoms, which interrupts or distorts the wavetrain emitted by a single system so as to widen its spread of Fourier components. There is a very large literature on these effects, whose detailed analysis is both taxing and controversial. No attempt will be made here to go beyond an elementary discussion and illustration of some of the leading ideas, with examples of how the line-width may be reduced or its effects mitigated for the purpose of high-precision measurements of the central frequency. We start with the second class of processes, and for our purpose the two-level system provides an adequate model, with ammonia as a practical realization.

The absorption spectrum of ammonia at a rather low pressure, 1.2 torr, in the wavelength range from 1.1 to 1.5 cm is shown in fig. 18.8, each line resulting from transitions between pairs of levels in different rotational states of the molecule, as defined by the pairs of quantum numbers above each line.

We begin with some examples of one-dimensional vibration in a nonparabolic potential chosen to permit complete analytical solution of Schrödinger's time-independent equation. These examples are of isochronous vibrators which classically have a frequency independent of amplitude, and which might be expected therefore to have energy levels equally spaced at a separation of ħω0. This expectation turns out to be very nearly right and inspires a certain confidence in the semi-classical procedure developed by (among others) Bohr, Wilson and Sommerfeld. We therefore apply this procedure to some non-isochronous systems and find once more rather good agreement with the results of exact quantum mechanics. Periodic systems in fact can often be treated semi-classically with adequate accuracy, and significant economy of effort in comparison with strict quantum-mechanical analysis. This approach pays handsomely when we turn in the next chapter to the quantization of electron cyclotron orbits which, as already discussed in chapter 8, are closely related to harmonic oscillators. Conduction electrons in semi-conductors, and still more in metals, have their behaviour modified by the lattice through which they move, and a complete quantal treatment has never been achieved. It is clear, however, from approximate calculations, often of great complexity, that the semiclassical method describes most of the interesting physical processes correctly and very simply. In chapter 15 we shall describe in outline some of the effects which can be treated quite well enough for most purposes without even writing down Schrödinger's equation.

Linear systems whose parameters are independent of time possess, as has been abundantly illustrated already, well-defined normal modes from which their motion can be synthesized by superposition; and the response to an applied force, varying with time, can be written in terms of the response to each separate Fourier component of the force. The same is not true of non-linear systems, since superposition is no longer a valid procedure for synthesizing the response. Every anharmonic system responds differently to a given form of time-dependent force, and even when the response has been found in any special case it will not scale up unchanged in response to an amplification of the force. Thus the response to a sinusoidal force is in general non-sinusoidal, the waveform changing with the amplitude of the force. There are very few general statements that can be made about the character of the response. One cannot even assert that the oscillations of the vibrator will have the same fundamental frequency as the applied force – it may respond at a subharmonic frequency, i.e. an integral submultiple of that of the force, or the response may be asynchronous to the point of randomness. Even when order prevails, with regular vibration at the fundamental or subharmonic frequency, changing the amplitude or frequency of the applied force to an infinitesimal degree may have the effect of throwing the response into an entirely new pattern.

A resonant system acted upon by an oscillatory force presents a straightforward enough problem if it is passive and linear, especially if the force is applied by some prime mover that is uninfluenced by the response it excites. Such problems are the subject matter of chapter 6, while nonlinear passive systems, as discussed in chapter 9, are more complicated to handle. If the prime mover is influenced by the response, additional complexities enter, and this chapter treats of some of these. As is to be expected, linear systems present the least difficulty, and we shall begin with the behaviour of two coupled resonant systems, each of which may be thought of as driving the other and being perturbed by the reaction of the other back on it. Examples have already appeared earlier, as for instance the coupled pendulums discussed in chapter 2, and the coupled resonant lines in chapter 7. In both cases we noted a very general characteristic of such systems, that even if they are tuned to the same frequency before being coupled, they do not vibrate at this frequency when coupled, but have resonances which move progressively further from the original frequency as the coupling is strengthened. It would perhaps be logical, having considered this problem, to proceed to coupled, passive, non-linear vibrators; but these are so difficult that we shall leave them alone. It is not quite so hard to derive useful results for the behaviour of self-maintained oscillators when perturbed either by the injection of a steady signal or by being coupled to a similar oscillator.

As is appropriate in an exposition of the quantum mechanics of vibrators, we started with a quantal calculation and have finished with one; but we have never strayed far from the classical models. We began, indeed, in chapter 13 the task of establishing limits to the validity of classical reasoning, and the very last example of this chapter has been used to demonstrate, somewhat perversely it might be thought, that the method of quantum mechanics can occasionally be applied to systems which physicists and engineers would instinctively regard as classical. So long as the discussions give insight into physical processes and reveal the strengths and weaknesses of the analytical tools available, no apology is needed and no defence is offered except the evidence of the book itself. Even the rather simple systems which provide material for the whole of the volume demonstrate clearly the increased power available to those who can handle both classical and quantal reasoning. The reader who wishes to apply his skill to complex vibratory processes will find his tasks eased if he can use whichever seems advantageous at each stage, and be confident that his understanding of simple processes will enable him to recognize the dangers inherent in both approaches – the danger of carrying classical reasoning too far into the quantum domain, and the danger of forcing over-simplifications on physical systems to make them amenable to the unforgiving methodology of quantum mechanics.