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Oscillations and waves are ubiquitous phenomena in the world around us. An oscillation is defined as a disturbance in a physical system that is repetitive in time. A wave is defined as a disturbance in a continuous, spatially extended, physical system that is both repetitive in time and periodic in space. In general, an oscillation involves a continual back and forth flow between two different energy types. For example, in the case of a pendulum, the two energy types are kinetic and gravitational potential energy. A wave involves similar repetitive energy flows to an oscillation, but, in addition, is capable of transmitting energy (and information) from one place to another. Although sound waves and electromagnetic waves, for example, rely on quite distinct physical mechanisms, they, nevertheless, share many common properties. This is also true of different types of oscillation. It turns out that the common factor linking the various types of wave and oscillation is that they are all described by the same mathematical equations.

The aim of this textbook is to develop a unified mathematical theory of oscillations and waves. Examples are drawn from the physics of discrete mechanical systems; continuous gases, fluids, and elastic solids; electronic circuits; electromagnetic waves; optical systems; and, finally, quantum mechanical systems.

It is assumed that readers of this book possess a basic familiarity with the laws of physics, such as might be obtained from a standard two-semester introductory college-level survey course. Readers are also assumed to be conversant with college-level mathematics up to and including algebra, trigonometry, linear algebra, ordinary differential equations, and partial differential equations.

One unusual feature of this textbook is that the introduction of the conventional complex representation of oscillations and waves is delayed until it becomes absolutely necessary (during the discussion of quantum mechanical waves). The reason for this choice is that, although the complex representation of oscillations and waves greatly facilitates calculations, it is (at least, initially) a significant obstacle to the development of a physical understanding of such phenomena. The author is of the opinion that students should first thoroughly understand how to represent oscillations and waves in terms of regular trigonometric functions before attempting to use the more convenient, but much more abstract, complex representation.

This book only deals with that class of oscillations and waves whose governing differential equations are linear. In most physical systems, this implies a restriction to relatively low amplitude phenomena. The author has resisted the temptation to discuss nonlinear oscillations and waves, mainly because such phenomena require a completely different sort of mathematical analysis to that used to describe linear oscillations and waves, and the main emphasis of this book is the mathematical unity of the subject matter.

Light is ultimately a wave phenomenon. Hence, it is natural that part of this book should be devoted to the study of optics. For the sake of brevity, however, only those aspects of optics that depend crucially on the wave-like nature of light (i.e., wave optics, rather than geometric optics) are discussed in any detail.

This textbook was developed for the ``Oscillations and Waves'' course that is currently taught at the University of Texas at Austin (UT) immediately following the standard mechanics/heat/sound and electricity/magnetism/light/atomic survey courses. The purpose of the UT waves course is to ease the difficult transition between lower-division physics courses, which mostly rely on algebraic equations, and upper-division courses, which rely almost exclusively on differential equations. Experience at UT indicates that the attrition of physics majors is particularly severe at this transition. On the other hand, experience also suggests that a lower-division waves course--which includes much more interesting applications of physics than the rather pedestrian applications that crop up in the aforementioned survey courses, while not requiring particularly advanced mathematics--is an effective means of converting undecided science students into physics majors.

The author would like to thank Alan Saeed for pointing out some errors in early drafts of this book.

next up previous
Next: Simple Harmonic Oscillation Up: Oscillations and Waves Previous: Oscillations and Waves
Richard Fitzpatrick 2013-04-08