Physical Sciences and Mathematics Commons^™

12 Cylindrical Coordinates, Charles G. Torre

Foundations of Wave Phenomena

We have seen how to build solutions to the wave equation by superimposing plane waves with various choices for amplitude, phase and wave vector k. In this way we can build up solutions which need not have the plane symmetry (exercise), or any symmetry whatsoever. Still, as you know by now, many problems in physics are fruitfully analyzed when they are modeled as having various symmetries, such as cylindrical symmetry or spherical symmetry. For example, the magnetic field of a long, straight wire carrying a steady current can be modeled as having cylindrical symmetry. Likewise, the sound waves emitted by …

Problem Set 2, Charles G. Torre

Foundations of Wave Phenomena

Problem Set 2

08 Fourier Analysis, Charles G. Torre

Foundations of Wave Phenomena

We now would like to show that one can build up the general solution of the wave equation by superimposing certain elementary solutions. Indeed, the elementary solutions being referred to are those discussed in §6. These elementary solutions will form a very convenient “basis” for the vector space of solutions to the wave equation, just as the normal modes provided a basis for the space of solutions in the case of coupled oscillators. Indeed, as we shall see, the elementary solutions are the normal modes for wave propagation. The principal tools needed to understand this are provided by the methods …

17 Maxwell Equations, Charles G. Torre

Foundations of Wave Phenomena

With our brief review of vector analysis out of the way, we can now discuss the Maxwell equations. We use the Gaussian system of electromagnetic units and let c denote the speed of light in vacuum. The Maxwell equations are differential equations for the electric field E(r, t), and the magnetic field B(r, t), which are defined by the force they exert on a test charge q at the point r at time t. This force is defined by the Lorentz force law.

15 Schrodinger Equation, Charles G. Torre

Foundations of Wave Phenomena

An important feature of the wave equation is that its solutions q(r, t) are uniquely specified once the initial values q(r, 0) and (del)q(r, 0)/@t are specified. As was mentioned before, if we view the wave equation as describing a continuum limit of a network of coupled oscillators, then this result is very reasonable since one must specify the initial position and velocity of an oscillator to uniquely determine its motion. It is possible to write down other “equations of motion” that exhibit wave phenomena but which only require the initial values of the dynamical variable — not its time …

18 The Electromagnetic Wave Equation, Charles G. Torre

Foundations of Wave Phenomena

Let us now see how the Maxwell equations (17.2)–(17.5) predict the existence of electromagnetic waves. For simplicity we will consider a region of space and time in which there are no sources (i.e., we consider the propagation of electromagnetic waves in vacuum). Thus we set p = 0 = j in our space-time region of interest. Now all the Maxwell equations are linear, homogeneous.

Problem Set 5, Charles G. Torre

Foundations of Wave Phenomena

Problem Set 5

Problem Set 7, Charles G. Torre

Foundations of Wave Phenomena

Problem Set 7

Vector Spaces (Appendix B), Charles G. Torre

Foundations of Wave Phenomena

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.

20 Polarization, Charles G. Torre

Foundations of Wave Phenomena

Our final topic in this brief study of electromagnetic waves concerns the phenomenon of polarization, which occurs thanks to the vector nature of the waves. More precisely, the polarization of an electromagnetic plane wave concerns the direction of the electric (and magnetic) vector fields. Let us first give a rough, qualitative motivation for the phenomenon. An electromagnetic plane wave is a traveling sinusoidal disturbance in the electric and magnetic fields. Let us focus on the behavior of the electric field since we can always reconstruct the behavior of the magnetic field from the electric field. Because the electric force on …

Problem Set 1, Charles G. Torre

Foundations of Wave Phenomena

Problem Set 1

Problem Set 10, Charles G. Torre

Foundations of Wave Phenomena

Problem Set 10

Problem Set 8, Charles G. Torre

Foundations of Wave Phenomena

Problem Set 8

01 Harmonic Oscillations, Charles G. Torre

Foundations of Wave Phenomena

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.

Read Me, Charles G. Torre

Foundations of Wave Phenomena

What this book is all about, why it was written, and stuff like that.

21 Non-Linear Wave Equations And Solitons, Charles G. Torre

Foundations of Wave Phenomena

In 1834 the Scottish engineer John Scott Russell observed at the Union Canal at Hermiston a well-localized* and unusually stable disturbance in the water that propagated for miles virtually unchanged. The disturbance was stimulated by the sudden stopping of a boat on the canal. He called it a “wave of translation”; we call it a solitary wave. As it happens, a number of relatively complicated – indeed, non-linear – wave equations can exhibit such a phenomenon. Moreover, these solitary wave disturbances will often be stable in the sense that if two or more solitary waves collide after the collision they …

19 Electromagnetic Energy, Charles G. Torre

Foundations of Wave Phenomena

In a previous physics course you should have encountered the interesting notion that the electromagnetic field carries energy and momentum. If you have ever been sunburned, you have experimental confirmation of this fact! We are now in a position to explore this idea quantitatively. In physics, the notions of energy and momentum are of interest mainly because they are conserved quantities. We can uncover the energy and momentum quantities associated with the electromagnetic field by searching for conservation laws. As before, such conservation laws will appear embodied in a continuity equation. Thus we begin by investigating a continuity equation for …

16 The Curl, Charles G. Torre

Foundations of Wave Phenomena

In §17–§20 we will study the mathematical basics behind the propagation of light waves, radio waves, microwaves, etc. All of these are, of course, examples of electromagnetic waves, that is, they are all the same (electromagnetic) phenomena just differing in their wavelength. The (non-quantum) description of all electromagnetic phenomena is provided by the Maxwell equations. These equations are normally presented as differential equations for the electric field E(r, t) and the magnetic field B(r, t). You may have been first introduced to them in an equivalent integral form. In differential form, the Maxwell equations involve the divergence operation, which we …

10 Why "Plane" Waves?, Charles G. Torre

Foundations of Wave Phenomena

Let us now pause to explain in more detail why we called the elementary solutions (9.9) and (9.26) plane waves. The reason is that the displacement q(r, t) has the symmetry of a plane. To see this, fix a time t (take a “snapshot” of the wave) and pick a location r. Examine the wave displacement q (at the fixed time) at all points in a plane that is (i) perpendicular to k, and (2) passes through r. The wave displacement will be the same at each point of this plane.

07 General Solution Of The One-Dimensional Wave Equation, Charles G. Torre

Foundations of Wave Phenomena

We will now find the “general solution” to the one-dimensional wave equation (5.11). What this means is that we will find a formula involving some “data” — some arbitrary functions — which provides every possible solution to the wave equation.

06 Elementary Solutions To The Wave Equation, Charles G. Torre

Foundations of Wave Phenomena

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.

05 The Continuum Limit And The Wave Equation, Charles G. Torre

Foundations of Wave Phenomena

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 …

Taylor’S Theorem And Taylor Series (Appendix A), Charles G. Torre

Foundations of Wave Phenomena

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.

Problem Set 3, Charles G. Torre

Foundations of Wave Phenomena

Problem Set 3

Problem Set 6, Charles G. Torre

Foundations of Wave Phenomena

Problem Set 6

References And Suggestions For Further Reading (Appendix C), Charles G. Torre

Foundations of Wave Phenomena

References and Suggestions for Further Reading (Appendix C)

Problem Set 9, Charles G. Torre

Foundations of Wave Phenomena

Problem Set 9

Pathway Pioneer: Heterogeneous Server Architecture For Scientific Visualization And Pathway Search In Metabolic Network Using Informed Search, Vipul Kantilal Oswal

All Graduate Theses and Dissertations, Spring 1920 to Summer 2023

PathwayPioneer is a website to visualize the biological models. The complex metabolic models are hard to perceive when read through text format. It is even more difficult to understand the flow of network and different subsystem or pathways. PathwayPioneer organizes different subsystems into simple graphical representation. We simplify the complex network based on subsystem and compartment. Users can do computation (flux balance analysis) through the user interface and perceive the changed model visually. Our system is one of its kind to handle large model files with quick response time.

Biologists are often interested in knowing different alternative pathways targeting a …

Multi-Scale Modeling Of Microbial Defection In The Presence Of Antibiotics, Darshan Dilip Nahar

All Graduate Theses and Dissertations, Spring 1920 to Summer 2023

Limited environmental resources push life into competition. In terms of microbes, we assume that the only goal is to grow and competition means to grow faster than the peers. Organisms tend to adapt to their environmental conditions and improve their efficiency in using the common resources to grow faster. While different organisms compete they engage into various social behaviors like cooperation to work together towards a common goal, e.g., production of public goods that benefits the entire population. Hence a part of the resources at hand is dedicated to participate in cooperative activities if the cost-to-benefit ratio is less than …

14 Conservation Of Energy, Charles G. Torre

Foundations of Wave Phenomena

After all of these developments it is nice to keep in mind the idea that the wave equation describes (a continuum limit of) a network of coupled oscillators. This raises an interesting question. Certainly you have seen by now how important energy and momentum — and their conservation — are for understanding the behavior of dynamical systems such as an oscillator. If a wave is essentially the collective motion of many oscillators, might not there be a notion of conserved energy and momentum for waves? If you’ve ever been to the beach and swam in the ocean you know that …