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Foundations Of Wave Phenomena: Complete Version, Charles G. Torre

Foundations of Wave Phenomena

This is the complete version of Foundations of Wave Phenomena. Version 8.3.1.

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Introduction To Classical Field Theory, Charles G. Torre

All Complete Monographs

This is an introduction to classical field theory. Topics treated include: Klein-Gordon field, electromagnetic field, scalar electrodynamics, Dirac field, Yang-Mills field, gravitational field, Noether theorems relating symmetries and conservation laws, spontaneous symmetry breaking, Lagrangian and Hamiltonian formalisms.

Scattering Of Light From Periodic Conducting Nanostructures, Wesley Mills

Fall Student Research Symposium 2021

Light scattered from periodic structures generates numerous fascinating phenomena, from diffraction patterns to the black patches on some butterfly wings. The simple scalar wave formulation based on Huygens principle can account for many diffraction and radiation patterns in the far field. However, when the structure dimensions are smaller than the wavelength, light polarization and structure details become important, and the vector formulation based on Maxwell's equations is necessary. I will present an analytic calculation and a numerical simulation on light scattering from two-dimensional conducting grids to model the reflectance from butterfly wings and broadband absorption structures made from carbon nanotube …

The Ideal Dog, Owen Graham

Fall Student Research Symposium 2020

The goal of this project was to look into all of the different jobs or tasks that we as humans have dogs perform and try to pinpoint what exactly it is that makes a dog ideal for each task. After identifying desirable characteristics, I considered physical traits in order to create an ideal dog that would be able to perform the greatest number of jobs possible.

National Security & Climate Change, Madison Mortensen

Fall Student Research Symposium 2020

Certain scientific subjects are often divisive or technical, which makes those topics difficult to discuss with audiences outside the scientific sphere. One way of getting around this obstacle is to cater scientific communication to different target audiences to cut through any audience biases. In order to accomplish that, a communicator needs to understand the relationship between audiences' worldviews, and what they know, feel, and do regarding the subject at hand, and then how that relationship influences the types of media audiences trust and to which they respond positively. The following study investigates the worldviews of a military audience with respect …

Galactic Sources In Gamma-Ray Diffuse Emission, Melissa Rasmussen

Fall Student Research Symposium 2020

The Fermi Large Area Telescope (LAT) has detected hundreds of Galactic gamma-ray sources, most of them pulsars. But the Galaxy contains tens of thousands of such sources which are still undetected due to their low flux, or because of conflation of the foreground with sources. Characterizing the general properties of detected sources would allow us to estimate the contribution to the diffuse Galactic emission from these undetected sources and in turn it would help the detection of new sources and even searches for dark matter. We present updates on our long-term effort to characterize the general properties of Galactic gamma-ray …

Dog-Headed: Outcast To All, Wesley Mills

Fall Student Research Symposium 2020

Cynocephali are a group of mythological, dog-headed, humans that appear throughout various cultures. Despite the wide range of region and time period in which they are used, they consistently represent a quality of "otherness".

Modeling Reflectance Spectra Of Nanorod Arrays With Arrays Of Light Sources, Christian Lange

Fall Student Research Symposium 2020

It is known that carbon-nanotube forests, nanopillar arrays, and other formations of quasi-periodic nanostructures of various materials (semiconductors, semimetals, and metals) can display a very low light reflectance over a wide range of wavelengths, and that the reflectance eventually starts to rise beyond an onset wavelength. As these materials can be quite reflective in planar form, this phenomenon indicates that morphology rather than material plays a dominant role. However, a quantitative analysis of the reflectance spectra of periodic structures has yet to be established. As a first step, we use an array of light sources to model the reflection from …

03 How To Find Normal Modes, Charles G. Torre

Foundations of Wave Phenomena

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.

02 Coupled Oscillators, Charles G. Torre

Foundations of Wave Phenomena

Our next step on the road to a bona fide wave is to consider a more interesting oscillating system: two coupled oscillators.

11 Separation Of Variables, Charles G. Torre

Foundations of Wave Phenomena

There is yet another way to find the general solution to the wave equation which is valid in 1, 2, or 3 (or more!) dimensions. This method is quite important and, as we shall see, can often be used for other linear homogeneous differential equations. This technique for solving the wave equation is called the method of separation of variables.

04 Linear Chain Of Coupled Oscillators, Charles G. Torre

Foundations of Wave Phenomena

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 …

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 …