Distributed loads

The gravitational force acting on a body is distributed over the whole volume of the body. However, if the body is rigid, we can replace this system of distributed gravitational forces by a single resultant gravitational force (weight) acting through the centre of gravity of the body.

It is helpful sometimes to represent a system of distributed parallel forces by a so-called load diagram. Consider a horizontal beam AB with sand piled on it to a constant depth d, as illustrated in Figure 4.1. We can then represent the load due to the sand and the weight of the beam by a load diagram with constant intensity q, as shown in Figure 4.2. If the total weight of the sand over the beam is S, the weight of the beam is W and the length of the beam is a, then qa = S + W. Measuring distance in metres and force in newtons, the intensity q has units N/m.

If the sand is heaped with a varying depth d, as shown in Figure 4.3a, then the corresponding load diagram, shown in Figure 4.3b, reduces at the ends A and B to an intensity equal to the weight per unit length of the beam. As in the case of uniform loading, the magnitude of the resultant or total load Q = S + W must equal the area of the load diagram ABDC.

Method of sections

A truss is a vertical framework of struts connected together at their ends so as to form a rigid structure, even when the connections are smooth hinge points. We shall assume that the structure is light compared with any supported loads. Thus, in the truss shown in Figure 5.1, we will neglect the weight of the struts which is assumed to be small compared with the load L. Furthermore, all the joints will be regarded as hinge points, i.e. any moments exerted at the joints are small enough to be neglected. This means that a strut will exert a force at a joint in the direction of the length of the strut. This force will be a push if the strut is in compression or a pull if it is in tension.

Finally, we shall only consider trusses with no redundant struts. In other words, the truss would collapse under the action of the load if any one strut were removed. Such a structure may be built up as a series of triangles. The struts need not have the same length, so the triangles need not be equilateral, as in Figure 5.1. However, there is a relation between the number s of struts and the number j of joints. For one triangle, s = j = 3. Then for each triangle added after that, there are two extra struts and one extra joint. It follows that s = 2j − 3.

Fraunhofer diffraction

The application of Fourier theory to Fraunhofer diffraction problems and to interference phenomena generally, was hardly recognized before the late 1950s. Consequently, only textbooks written since then mention the technique. Diffraction theory, of which interference is only a special case, derives from Huygens' principle: that every point on a wavefront which has come from a source can be regarded as a secondary source: and that all the wavefronts from all these secondary sources combine and interfere to form a new wavefront.

Some precision can be added by using calculus. In the diagram (Fig. 3.1), suppose that at O there is a source of ‘strength’ q, defined by the fact that at A, a distance r from O there is s ‘field’, E of strength E=q/r. Huygens' principle is now as follows:

In Fraunhofer diffraction we simplify. We assume:

• that only two dimensions need be considered. All apertures bounding the transparent part of the surface S are rectangular and of length unity perpendicular to the plane of the diagram.

• that the dimensions of the aperture are small compared with r′.

• that r is very large so that the field E has the same magnitude at all points on the transparent part of S, and a slowly varying or constant phase. (Another way of putting it is to say that plane wavefronts arrive at the surface S from a source at -∞).

• […]

Showing a Fourier transform to a physics student generally produces the same reaction as showing a crucifix to Count Dracula. This may be because the subject tends to be taught by theorists who themselves use Fourier methods to solve otherwise intractable differential equations. The result is often a heavy load of mathematical analysis.

This need not be so. Engineers and practical physicists use Fourier theory in quite another way: to treat experimental data, to extract information from noisy signals, to design electrical filters, to ‘clean’ TV pictures and for many similar practical tasks. The transforms are done digitally and there is a minimum of mathematics involved.

The chief tools of the trade are the theorems in Chapter 2, and an easy familiarity with these is the way to mastery of the subject. In spite of the forest of integration signs throughout the book there is in fact very little integration done and most of that is at high-school level. There are one or two excursions in places to show the breadth of power that the method can give. These are not pursued to any length but are intended to whet the appetite of those who want to follow more theoretical paths.

The book is deliberately incomplete. Many topics are missing and there is no attempt to explain everything: but I have left, here and there, what I hope are tempting clues to stimulate the reader into looking further; and of course, there is a bibliography at the end.

This edition follows much advice and constructive criticism which the author has received from all quarters of globe, in consequence of which various typos and misprints have been corrected and some ambiguous statements and anfractuosities have been replaced by more clear and direct derivations. Chapter 7 has been largely rewritten to demonstrate the way in which Fourier transforms are used in CAT-scanning, an application of more than usual ingenuity and importance: but overall this edition represents a renewed effort to rescue Fourier transforms from the clutches of the pure mathematicians and present them as a working tool to the horny-handed toilers who strive in the fields of electronic engineering and experimental physics.

The qualitative approach

Ninety percent of all physics is concerned with vibrations and waves of one sort or another. The same basic thread runs through most branches of physical science, from accoustics through engineering, fluid mechanics, optics, electromagnetic theory and X-rays to quantum mechanics and information theory. It is closely bound to the idea of a signal and its spectrum. To take a simple example: imagine an experiment in which a musician plays a steady note on a trumpet or a violin, and a microphone produces a voltage proportional to the the instantaneous air pressure. An oscilloscope will display a graph of pressure against time, F(t), which is periodic. The reciprocal of the period is the frequency of the note, 256 Hz, say, for a well-tempered middle C.

The waveform is not a pure sinusoid, and it would be boring and colourless if it were. It contains ‘harmonics’ or ‘overtones’: multiples of the fundamental frequency, with various amplitudes and in various phases, depending on the timbre of the note, the type of instrument being played and on the player. The waveform can be analysed to find the amplitudes of the overtones, and a list can be made of the amplitudes and phases of the sinusoids which it comprises. Alternatively a graph, A(v), can be plotted (the sound-spectrum) of the amplitudes against frequency.

History

Fourier transformation is formally an analytical process which uses integral calculus. In experimental physics and engineering, however, the integrand may be a set of experimental data, and the integration is necessarily done artificially. Since a separate integration is needed to give each point of the transformed function, the process would become exceedingly tedious if it were to be attempted manually and many ingenious devices have been invented for performing Fourier transforms mechanically, electrically, acoustically and optically. These are all now part of history since the arrival of the digital computer and more particularly since the discovery – or invention – of the ‘Fast Fourier Transform’ algorithm or FFT as it is generally called. Using this algorithm, the data are put (‘read’) into a file (or ‘array’, depending on the computer jargon in use); the transform is carried out, and the array then contains the points of the transformed function. It can be achieved by a software program, or by a purpose-built integrated circuit. It can be done very quickly so that vibration-sensitive instruments with Fourier transformers attached can be used for tuning pianos and motor engines, for aircraft and submarine detection and so on. It must not be forgotten that the ear is Nature's own Fourier transformer, and, as used by an expert piano-tuner for example, is probably the equal of any electronic simulator in the 20–20 000 Hz range.