# Physics Commons™

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## Full-Text Articles in Physics

Introduction To Classical Field Theory, Charles G. Torre Aug 2019

#### 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.

Foundations Of Wave Phenomena: Complete Version, Charles G. Torre Dec 2016

#### 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.

Please click here to explore the components of this work.

06 Elementary Solutions To The Wave Equation, Charles G. Torre Aug 2014

#### 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.

18 The Electromagnetic Wave Equation, Charles G. Torre Aug 2014

#### 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 8, Charles G. Torre Aug 2014

#### Problem Set 8, Charles G. Torre

##### Foundations of Wave Phenomena

Problem Set 8

11 Separation Of Variables, Charles G. Torre Aug 2014

#### 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.

12 Cylindrical Coordinates, Charles G. Torre Aug 2014

#### 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 ...

13 Spherical Coordinates, Charles G. Torre Aug 2014

#### 13 Spherical Coordinates, Charles G. Torre

##### Foundations of Wave Phenomena

The spherical coordinates of a point p can be obtained by the following geometric construction. The value of r represents the distance from the point p to the origin (which you can put wherever you like). The value of ✓ is the angle between the positive z-axis and a line l drawn from the origin to p. The value of " is the angle made with the x-axis by the projection of l into the x-y plane (z = 0). Note: for points in the x-y plane, r and " (not ✓) are polar coordinates. The coordinates (r, ✓, ") are called the radius, polar angle, and ...

04 Linear Chain Of Coupled Oscillators, Charles G. Torre Aug 2014

#### 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 left ...

Problem Set 5, Charles G. Torre Aug 2014

#### Problem Set 5, Charles G. Torre

##### Foundations of Wave Phenomena

Problem Set 5

References And Suggestions For Further Reading (Appendix C), Charles G. Torre Aug 2014

#### 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 6, Charles G. Torre Aug 2014

#### Problem Set 6, Charles G. Torre

##### Foundations of Wave Phenomena

Problem Set 6

Problem Set 10, Charles G. Torre Aug 2014

#### Problem Set 10, Charles G. Torre

##### Foundations of Wave Phenomena

Problem Set 10

20 Polarization, Charles G. Torre Aug 2014

#### 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 ...

05 The Continuum Limit And The Wave Equation, Charles G. Torre Aug 2014

#### 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 to ...

09 The Wave Equation In 3 Dimensions, Charles G. Torre Aug 2014

#### 09 The Wave Equation In 3 Dimensions, Charles G. Torre

##### Foundations of Wave Phenomena

We now turn to the 3-dimensional version of the wave equation, which can be used to describe a variety of wavelike phenomena, e.g., sound waves and electromagnetic waves. One could derive this version of the wave equation much as we did the one-dimensional version by generalizing our line of coupled oscillators to a 3-dimensional array of oscillators. For many purposes, e.g., modeling propagation of sound, this provides a useful discrete model of a three dimensional solid.

15 Schrodinger Equation, Charles G. Torre Aug 2014

#### 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 ...

08 Fourier Analysis, Charles G. Torre Aug 2014

#### 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 ...

02 Coupled Oscillators, Charles G. Torre Aug 2014

#### 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.

Taylor’S Theorem And Taylor Series (Appendix A), Charles G. Torre Aug 2014

#### 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 7, Charles G. Torre Aug 2014

#### Problem Set 7, Charles G. Torre

##### Foundations of Wave Phenomena

Problem Set 7

Problem Set 1, Charles G. Torre Aug 2014

#### Problem Set 1, Charles G. Torre

##### Foundations of Wave Phenomena

Problem Set 1

Problem Set 4, Charles G. Torre Aug 2014

#### Problem Set 4, Charles G. Torre

##### Foundations of Wave Phenomena

Problem Set 4

Read Me, Charles G. Torre Aug 2014

#### Read Me, Charles G. Torre

##### Foundations of Wave Phenomena

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

Problem Set 9, Charles G. Torre Aug 2014

#### Problem Set 9, Charles G. Torre

##### Foundations of Wave Phenomena

Problem Set 9

10 Why "Plane" Waves?, Charles G. Torre Aug 2014

#### 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.

03 How To Find Normal Modes, Charles G. Torre Aug 2014

#### 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.

07 General Solution Of The One-Dimensional Wave Equation, Charles G. Torre Aug 2014

#### 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.

Problem Set 3, Charles G. Torre Aug 2014

#### Problem Set 3, Charles G. Torre

##### Foundations of Wave Phenomena

Problem Set 3

Problem Set 2, Charles G. Torre Aug 2014

Problem Set 2