Foundations of Wave PhenomenaCopyright (c) 2016 Utah State University All rights reserved.
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Recent documents in Foundations of Wave Phenomenaen-usWed, 06 Jan 2016 03:09:01 PST3600Problem Set 9
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Charles G. TorreProblem Set 10
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Charles G. TorreReferences and Suggestions for Further Reading (Appendix C)
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References and Suggestions for Further Reading (Appendix C)
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Charles G. TorreProblem Set 8
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Charles G. TorreProblem Set 7
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Charles G. TorreProblem Set 6
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Charles G. TorreProblem Set 5
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Charles G. TorreProblem Set 3
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Charles G. TorreProblem Set 4
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Charles G. TorreProblem Set 1
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Charles G. TorreProblem Set 2
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Charles G. TorreRead Me
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What this book is all about, why it was written, and stuff like that.
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Charles G. TorreTaylor’s Theorem and Taylor Series (Appendix A)
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Taylor’s theorem and Taylor’s series constitute one of the more important tools used by mathematicians, physicists and engineers. They provides a means of approximating a function in terms of polynomials.
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Charles G. TorreVector Spaces (Appendix B)
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Throughout this text we have noted that various objects of interest form a vector space. Here we outline the basic structure of a vector space. You may find it useful to refer to this Appendix when you encounter this concept in the text.
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Charles G. Torre01 Harmonic Oscillations
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Everyone has seen waves on water, heard sound waves and seen light waves. But, what exactly is a wave? Of course, the goal of this course is to answer this question for you. But for now you can think of a wave as a traveling or oscillatory disturbance in some continuous medium (air, water, the electromagnetic field, etc.). As we shall see, waves can be viewed as a collective e↵ect resulting from a combination of many harmonic oscillations. So, to begin, we review the basics of harmonic motion.
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Charles G. Torre02 Coupled Oscillators
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Our next step on the road to a bona fide wave is to consider a more interesting oscillating system: two coupled oscillators.
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Charles G. Torre03 How To Find Normal Modes
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How do we find the normal modes and resonant frequencies without making a clever guess? Well, you can get a more complete explanation in an upper-level mechanics course, but the gist of the trick involves a little linear algebra. The idea is the same for any number of coupled oscillators, but let us stick to our example of two oscillators.
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Charles G. Torre04 Linear Chain of Coupled Oscillators
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As an important application and extension of the foregoing ideas, and to obtain a first glimpse of wave phenomena, we consider the following system. Suppose we have N identical particles of mass m in a line, with each particle bound to its neighbors by a Hooke’s law force, with “spring constant” k. Let us assume the particles can only be displaced in one-dimension; label the displacement from equilibrium for the jth particle by qj , j = 1, ...,N. Let us also assume that particle 1 is attached to particle 2 on the right and a rigid wall on the left, and that particle N is attached to particle N 1 on the left and another rigid wall on the right.
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Charles G. Torre05 The Continuum Limit and the Wave Equation
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Our example of a chain of oscillators is nice because it is easy to visualize such a system, namely, a chain of masses connected by springs. But the ideas of our example are far more useful than might appear from this one simple mechanical model. Indeed, many materials (including solids, liquids and gases) have some aspects of their physical response to (usually small) perturbations behaving just as if they were a bunch of coupled oscillators — at least to a first approximation. In a sense we will explore later, even the electromagnetic field behaves this way! This “harmonic oscillator” response to perturbations leads — in a continuum model — to the appearance of wave phenomena in the traditional sense. We caught a glimpse of this when we examined the normal modes for a chain of oscillators with various boundary conditions. Because the harmonic approximation is often a good first approximation to the behavior of systems near equilibrium, you can see why wave phenomena are so ubiquitous. The key di↵erence between a wave in some medium and the examples of §4 is that wave phenomena are typically associated with propagation media (stone, water, air, etc.) which are modeled as continuous rather than discrete. As mentioned earlier, our chain of oscillators in §4 can be viewed as a discrete model of a continuous (one-dimensional) material. We now want to introduce a phenomenological description of the material in which we ignore the atomic discreteness of matter.
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Charles G. Torre06 Elementary Solutions to the Wave Equation
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Before systematically exploring the wave equation, it is good to pause and contemplate some basic solutions. We are looking for a function q of 2 variables, x and t, whose second x derivatives and second t derivatives are proportional. You can probably guess such functions with a little thought. But our derivation of the equation from the model of a chain of oscillators gives a strong hint.
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Charles G. Torre