We now take a quick look at a few representative physical systems, so that we can appreciate the great variety of circumstances in which the mathematics of harmonic motion can be applied. The model system has taught us what properties we must look for if we wish to know whether vibration is possible in any given real system. The mathematics of the prototype can be used in a production-line spirit once we have isolated the ingredients (inertia and stiffness) necessary for vibration.

As well as appreciating the essential similarity of different kinds of vibration, we shall in this chapter be vitally interested in the numerical sizes of the physical quantities involved, and particularly in the values of the free vibration frequencies of the various systems. One of the most striking things we shall discover is the very wide spread of frequencies to be found.

Angular vibrations

The mass in the model system vibrates to and fro along a straight line. Some of the most familiar vibrations involve massive objects rotating back and forth under the influence of a return torque. We shall consider two kinds of return torque: first an elastic torque provided by a stiff suspension, and then one due to gravity.

Torsional vibrations. The system shown in fig. 2.1 is the exact rotational analogue of the model system. The coordinate ψ now measures the angular displacement of the mass from its equilibrium position.

Systems which can be described in terms of a single coordinate are exceptional, and we must expect the vibrational possibilities to become more numerous and more complex as the required number of coordinates increases.

Most of the new features can be revealed by thinking about a system with only two coordinates. Even a two-coordinate system can exhibit very complicated vibrational behaviour, but we shall find that there are two very simple basic motions, known as modes, and that any other possible vibration can be treated as a superposition of these.

In the first two sections of this chapter we shall discover, without undue rigour, the physical nature of the modes and the way in which the mathematics can be simplified by the choice of a special coordinate system. The third section outlines the technique for finding these ‘mode coordinates’ for a given system, and the technique is applied to a physical example in section 8.4. Only sections 8.1 and 8.2 are strictly necessary to enable you to understand the rest of the book.

Modes and mode coordinates

We start with the simple system shown in fig. 8.1. It consists of two identical masses, connected to each other and to two fixed points by a symmetric arrangement of three springs, of which the outer ones each have stiffness s, and the central one has stiffness S. We ignore all forces other than the spring forces, and we consider one-dimensional motion of each mass along the line of the springs.

It would be generally agreed that an undergraduate course on vibrations and waves should lead the student to a thorough understanding of the basic concepts, demonstrate how these concepts unify a wide variety of familiar physics, and open doors to some more advanced topics on which they shed light. The fundamental ideas can be introduced with reference to mechanical systems which are easy to visualize and to illustrate; less tangible phenomena can then be treated with the same mathematics, leaving one free to concentrate on more interesting problems of formulation and interpretation.

In such a course there is always a risk that springs and strings will take over completely, submerging the real physics. The theory must be developed with some care and thoroughness if the student is to comprehend it in sufficient depth to be able to apply it readily, and all too often the physics has to be left half baked. This textbook is an attempt to provide, in a volume of moderate size, an account which is systematic and coherent, but which also treats the physical examples in some depth.

The diagram on p. viii shows the plan of the book, and indicates the relative weights attached to fundamental and illustrative material. The theoretical development (indicated by the downward flow through the toned areas on the left) is continuous, but is punctuated by regular excursions into physics.

The boundaries between different wave-carrying media discussed in the previous chapter were assumed to be extensive. Reflected and refracted waves generated at such boundaries have a family resemblance to the incident wave giving rise to them; plane waves arriving at plane boundaries, for example, result in plane reflected and refracted waves.

If the second medium occupies only a small pocket of space embedded within the first medium, we think of it less as a medium and more as an object obstructing the progress of the wave through the medium proper. A new type of behaviour, known as diffraction, is apparent in such cases. Particularly striking effects are obtained when large numbers of identical diffracting objects are arranged in some regular way. A familiar example is provided by a street lamp as seen through a fog of water particles: the halo round the lamp would not be there if the lamp were viewed through a continuous mass of water.

The most characteristic feature of diffraction is that incident waves appear to ‘bend round’ objects made of materials which, in the form of extended media, are quite opaque. In a similar way, waves passing through a transparent hole in an opaque object are ‘spread out’. We shall refer to an obstacle in a transparent medium, or an aperture in an opaque object, as a diffraction centre.

We shall find that these effects are appreciable when the diffraction centres have dimensions which are comparable with the wavelength of the waves involved.

Deriving the Lorentz transformations

Relativity theory is provocative. It incites an urge to show that the theory is inconsistent or that one can beat the speed of light. In this chapter we examine both issues. As a prelude, we need to study how Alice and Bob assign coordinates to an event, that is, how they specify the location and time of an event. Then we need to learn how those assignments are related. Once we have those relations, we can compute how differently Alice and Bob perceive the velocity of some object. And that is a major step toward trying to beat the speed of light.

Alice and Bob specify the location of an event – a firecracker going off, say – by measuring distances from the origins of their respective reference frames. Figure 13.1 displays this. The distance Bob assigns is denoted x; the distance Alice assigns is called x. Alice specifies the time by using a clock that is present at the event, is at rest in her frame, and is synchronized with the master clock at the origin of her frame. Bob does the same, using a clock at rest in his frame. This procedure – using clocks at the event itself – is a matter of convenience and is something we discussed earlier, in section 9.5. The times are denoted t for Bob's value and t' for Alice's.

Hertz revisited

In chapters 5 and 6, we noted that Heinrich Hertz discovered the photoelectric effect while investigating electromagnetic waves. A strong spark produces not only visible light but also ultraviolet light. When ultraviolet light from a spark in Hertz's transmitter circuitry struck the metal of his detector, the ultraviolet ejected some electrons. Those electrons, now present in the gap across which a spark was to jump, made it easier for a spark to jump. After all, a spark is an electric current, and the presence of the ejected electrons gave the spark a head start (for air is ordinarily a poor conductor of electricity, having few mobile charged particles). What Hertz perceived directly was that, when ultraviolet light shone on the detector, the spark was easier to start and could be made to jump a larger gap.

This was fascinating, and Hertz did spend six months establishing conclusively the essential role of ultraviolet light, but then he returned to electromagnetic waves.

First observations

How should we begin? Surely with Isaac Newton. He published his first scientific paper in 1672, and it was on light. In response to an objection raised against his paper, he wrote

• For the best and safest method of philosophizing seems to be, first diligently to investigate the properties of things and establish them by experiment, and then to seek hypotheses to explain them. For hypotheses ought to be fitted merely to explain the properties of things and not attempt to predetermine them, except in so far as they can be an aid to experiments.

So let us get a light source and begin investigating. I will suppose that you have access to some apparatus – at least in the sense of seeing lecture demonstrations – and so I will describe things as though we were doing the experiments together.

We take a strong flashlight or a 35-millimeter slide projector and shine the light horizontally. Clapping a pair of well-used erasers produces a cloud of chalk dust, and we see the light beam piercing the cloud as a bright shaft of light. This gives us our first observation:

Observation 1. Light goes in a straight line from a luminous source.

Reflection and refraction

Next we tip our source – the projector, say – so that the light enters the water in an aquarium tank. Figure 1.1 depicts what happens.

In chapter 11, I noted that the equation E=mc2 has given rise to a host of misconceptions. There are other issues associated with the equation, too, among them an issue of definition, and this appendix addresses some of those issues. It is written primarily for an instructor, but a student who has read appendix B will be able to understand most of it and will gain a greater perspective.

Mass in history

The word “mass” has a long history and a correspondingly long string of meanings and nuances. Max Jammer provides a fine discourse on the topic in his Concepts of Mass in'Classical and Modern Physics, cited in the “Additional resources” for chapter 11; I rely on Professor Jammer for the early history. In Roman times, the Latin word “massa” denoted a lump of dough. This sense of “lump of stuff” is what we have in mind when we “hang a mass from a spring.” By Newton's time, the word “massa” (still in Latin, of course) had come to denote “quantity of matter,” a valuable concept but one that defied attempts to make it both operationally precise and independent of any prior notion of density.

Newton recognized clearly the physical notion of inertia and hence what we would today call “inertial mass.” It was Leonhard Euler, however, who first defined inertial mass operationally as the ratio of force to acceleration (more precisely, as the ratio of their magnitudes).

Preview

The last three chapters developed a wave theory of light. Indeed, they suggested that light is an electromagnetic wave, and then they marshaled experimental support for that view. The wave theory of light is correct – but it is not the whole story. Some other idea complements that theory. In this chapter, new experiments compel us to develop the alternative idea, and in chapter 7 we learn how the wave theory and the new idea complement each other.

The photoelectric effect

In this day and age, we are accustomed to the idea that sunlight is a source of energy. Let us investigate the connection between light (in general) and energy.

If you are not thoroughly familiar with how the word “energy” is used in physics, now is the right time to read appendix A, Energy. Knowing what physicists mean with their words will help you to understand the line of reasoning.

Electron escape

When an electron absorbs light, it acquires additional energy. If the electron is in a piece of metal, it may be able to escape from the metal. The positive charges in the metal tend to hold the electrons back and prevent their escape. (To be sure, the electrons can move about quite freely inside a good electrical conductor.)

The electric field

Let us recall the plan we adopted at the close of section 3.1:

1. (a) Learn about waves.

2. (b) Find that light does indeed have a wave-like character.

3. (c) Learn what the “something” is out of which light waves are formed.

In subsequent sections, we found that waves do have the properties of reflection and refraction that light possesses. Then we shifted perspective and found that light has the property of interference that waves possess. In short, chapters 3 and 4 taught us that light does indeed have a wave-like character.

Before we can address part (c) of the plan, we need to know certain aspects of electricity and magnetism. So we turn now to those twin subjects.

Sometimes, when you pull a sweater off over your head on a day when the air is particularly dry (perhaps indoors in winter), you hear a crackling sound and may even feel an electric spark. The frictional rubbing of the sweater on your shirt or blouse separates some electric charges, one from another; the crackling and the spark are a miniature form of thunder and lightning.

More deliberate rubbing of two different objects separates electric charges more effectively.

Chapter 9 began with the assertion, “Space and time are different from what you thought they were like.” By now, probably you agree. This brief chapter draws together verbally the remarkable properties of the space and time in which we live.

The interconnection of space and time

Space and time are intrinsically different from each other. We determine locations in space by laying off a wooden meter stick or by stretching a metal tape, but we determine “locations in time” by reading a clock, be it a pendulum clock or a digital watch or a cesium beam atomic clock. Such an intrinsic difference was appreciated before Einstein's 1905 relativity paper, and it persists to this day. Nonetheless, relativity theory showed space and time to be more closely linked than people had thought. Let us review the connections.

The linkage of space and time shows up most clearly when Alice and Bob – in motion relative to each other – compare the locations in space and time that they assign to the same events. Our investigation of the relativity of simultaneity provides an excellent context. Look again at figures 9.6 and 9.7. Focus first on the event of light's reaching marker (a), fixed in Alice's frame. For Bob, the location in space of that event depends on both the distance of marker (a) from the origin of Alice's frame and on how long she has been moving since the origins of the two frames passed each other.

The college catalog said

• Physics 104. Newton to Einstein: The Trail of Light

• The course will follow the trail of light from Newton's corpuscles to Einstein's relativity. The major theoretical landmarks will be the wave-particle duality and the special theory of relativity.

It was the spring term of 1987, and the course was brand-new. Out of my previous experience with courses for non-scientists, I expected a class of 20 to 30 students. If enrollment became a fad, then the class might grow to 50 or 60. I was not prepared for what happened. When pre-registration was over, 141 students had signed up.

As an aside, let me remark that I was also scheduled to teach the calculusbased introductory course, ordinarily our largest course. That spring I taught more students than the rest of the department, all put together.

My intention was to teach the course in alternate years (to maximize the number of times I would teach it before burning out). When I taught the course the second time, in the fall of 1988, over 170 students sought admission. Reluctantly, I held the line at 150, my saturation point when reading essays and correcting exams.

During the last week of the semester, I ask my students for advice about improving the course, and I also ask them, “What, for you, is the most valuable thing you learned or ‘got’ from this course?” In 1988, my younger son, then a senior in high school and ever skeptical of his father's teaching methods, happened to read the responses first.