Vibrations and Waves

A.P. French

• Publisher: CRC Press
• Year: 2017
• ISBN:
  9781351989893
• Edition: 1
• Topic: Scienze fisiche
• Subtopic: Materia condensata

The M.I.T. Introductory Physics Series is the result of a program of careful study, planning, and development that began in 1960. The Education Research Center at the Massachusetts Institute of Technology (formerly the Science Teaching Center) was established to study the process of instruction, aids thereto, and the learning process itself, with special reference to science teaching at the university level. Generous support from a number of foundations provided the means for assembling and maintaining an experienced staff to co-operate with members of the Institute's Physics Department in the examination, improvement, and development of physics curriculum materials for students planning careers in the sciences. After careful analysis of objectives and the problems involved, preliminary versions of textbooks were prepared, tested through classroom use at M.I.T. and other institutions, re-evaluated, rewritten, and tried again. Only then were the final manuscripts undertaken.

1
Periodic motions

THE VIBRATIONS or oscillations of mechanical systems constitute one of the most important fields of study in all physics. Virtually every system possesses the capability for vibration, and most systems can vibrate freely in a large variety of ways. Broadly speaking, the predominant natural vibrations of small objects are likely to be rapid, and those of large objects are likely to be slow. A mosquito’s wings, for example, vibrate hundreds of times per second and produce an audible note. The whole earth, after being jolted by an earthquake, may continue to vibrate at the rate of about one oscillation per hour. The human body itself is a treasure-house of vibratory phenomena ; as one writer has put it¹:

After all, our hearts beat, our lungs oscillate, we shiver when we are cold, we sometimes snore, we can hear and speak because our eardrums and larynges vibrate. The light waves which permit us to see entail vibration. We move by oscillating our legs. We cannot even say “vibration” properly without the tip of the tongue oscillating… Even the atoms of which we are constituted vibrate.

The feature that all such phenomena have in common is periodicity. There is a pattern of movement or displacement that repeats itself over and over again. This pattern may be simple or complicated ; Fig. 1-1 shows an example of each—the rather complex cycle of pressure variations inside the heart of a cat, and the almost pure sine curve of the vibrations of a tuning fork. In each case the horizontal axis represents the steady advance of time, and we can identify the length of time—the period Τ—within which one complete cycle of the vibration is performed.

¹Frorn R. E. D. Bishop, Vibration, Cambridge University Press, New York, 1965. A most lively and fascinating general account of vibrations with particular reference to engineering problems.

Fig. 1-1 (a) Pressure variations inside the heart of a cat {After Straub, in E. H. Starling, Elements of Human Physiology, Churchill, London, 1907.)
(b) Vibrations of a tuning fork.

In this book we shall study a number of aspects of periodic motions, and will proceed from there to the closely related phenomenon of progressive waves. We shall begin with some discussion of the purely kinematic description of vibrations. Later, we shall go into some of the dynamical properties of vibrating systems—those dynamical features that allow us to see oscillatory motion as a real physical problem, not just as a mathematical exercise.

Sinusoidal Vibrations

Our attention will be directed overwhelmingly to sinusoidal vibrations of the sort exemplified by Fig. 1-1 (b). There are two reasons for this—one physical, one mathematical, and both basic to the whole subject. The physical reason is that purely sinusoidal vibrations do, in fact, arise in an immense variety of mechanical systems, being due to restoring forces that are proportional to the displacement from equilibrium. Such motion is almost always possible if the displacements are small enough. If, for example, we have a body attached to a spring, the force exerted on it at a

The description of simple harmonic motion

F(x)=−(k1x+k2x2+k3x3+⋅⋅⋅)

where k₁, k₂, k_3, etc., are a set of constants, and we can always find a range of values of x within which t...