
01 Harmonic Oscillations
Charles G. Torre
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.

02 Coupled Oscillators
Charles G. Torre
Our next step on the road to a bona fide wave is to consider a more interesting oscillating system: two coupled oscillators.

03 How To Find Normal Modes
Charles G. Torre
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 upperlevel 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.

04 Linear Chain of Coupled Oscillators
Charles G. Torre
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 onedimension; 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.

05 The Continuum Limit and the Wave Equation
Charles G. Torre
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 (onedimensional) material. We now want to introduce a phenomenological description of the material in which we ignore the atomic discreteness of matter.

06 Elementary Solutions to the Wave Equation
Charles G. Torre
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.

07 General Solution of the OneDimensional Wave Equation
Charles G. Torre
We will now find the “general solution” to the onedimensional 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.

08 Fourier Analysis
Charles G. Torre
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 of Fourier analysis, which is very useful in analyzing waves in any number of spatial dimensions. To begin, we will take a somewhat superficial tour of the key results of Fourier analysis. Then we’ll see how to use these results to better understand the solutions to the wave equation.

09 The Wave Equation in 3 Dimensions
Charles G. Torre
We now turn to the 3dimensional 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 onedimensional version by generalizing our line of coupled oscillators to a 3dimensional array of oscillators. For many purposes, e.g., modeling propagation of sound, this provides a useful discrete model of a three dimensional solid.

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

11 Separation of Variables
Charles G. Torre
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
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 a pointlike source are nicely approximated as spherically symmetric. Now, using the Fourier expansion in plane waves we can construct such symmetric solutions — indeed, we can construct any solution to the wave equation. But, as you also know, we have coordinate systems that are adapted to a variety of symmetries, e.g., cylindrical coordinates, spherical polar coordinates, etc. When looking for waves with some chosen symmetry it is advantageous to get at the solutions to the wave equation directly in these coordinates, without having to express them as a superposition of plane waves. Our task now is to see how to express solutions of the wave equations in a useful fashion in terms of such curvilinear coordinate systems.

13 Spherical Coordinates
Charles G. Torre
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 zaxis and a line l drawn from the origin to p. The value of " is the angle made with the xaxis by the projection of l into the xy plane (z = 0). Note: for points in the xy plane, r and " (not ✓) are polar coordinates. The coordinates (r, ✓, ") are called the radius, polar angle, and azimuthal angle of the point p, respectively. It should be clear why these coordinates are called spherical. The points r = a, with a = constant, lie on a sphere of radius a about the origin. Note that the angular coordinates can thus be viewed as coordinates on a sphere. Indeed, they label latitude and longitude.

14 Conservation of Energy
Charles G. Torre
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 waves do indeed carry energy and momentum which can be transferred to other systems. How to see energy and momentum and their conservation laws emerge from the wave equation? One way to answer this question would be to go back to the system of coupled oscillators and try to add up the energy and momentum of each oscillator at a given time and take the continuum limit to get the total energy and momentum of the wave. Of course, the energy and momentum of each individual oscillator is not conserved (exercise). Indeed, the propagation of a wave depends upon the fact that the oscillators are coupled, i.e., can exchange energy and momentum. What we want to do here, however, is to show how to keep track of this energy flow in a wave, directly from the continuum description we have been developing. This will allow us to define the energy (and momentum) densities of the wave as well as the total energy contained in a region.

15 Schrodinger Equation
Charles G. Torre
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 derivative — to specify a solution. This is physically appropriate in a number of situations, the most significant of which is in quantum mechanics where the wave equation is called the Schrodinger equation. This equation describes the time development of the observable attributes of a particle via the wave function (or probability amplitude) . In quantum mechanics, the complete specification of the initial conditions of the particle’s motion is embodied in the initial value of . The price paid for this change in the allowed initial data while asking for a linear wave equation is the introduction of complex numbers into the equation for the wave. Indeed, the values taken by are complex numbers. In what follows we shall explore some of the elementary features of the wave phenomena associated with the Schrodinger equation.

16 The Curl
Charles G. Torre
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 (nonquantum) 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 mentioned before, and another vector differential operator, known as the curl. In preparation for our discussion of electromagnetic waves, we explore this vector differential operator in a little detail.

17 Maxwell Equations
Charles G. Torre
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.

18 The Electromagnetic Wave Equation
Charles G. Torre
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 spacetime region of interest. Now all the Maxwell equations are linear, homogeneous.

19 Electromagnetic Energy
Charles G. Torre
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 energy and momentum. As was mentioned earlier, there are systematic methods to search for continuity equations associated with a system of differential equations such as the Maxwell equations, but we will not get into that. Instead, we simply write down the continuity equation associated to energymomentum conservation.

20 Polarization
Charles G. Torre
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 a charged particle is along the direction of the electric field, the response of charges to electromagnetic waves is sensitive to the direction of the electric field in a plane wave. Such effects are what we refer to when we discuss polarization phenomena involving light.

21 NonLinear Wave Equations and Solitons
Charles G. Torre
In 1834 the Scottish engineer John Scott Russell observed at the Union Canal at Hermiston a welllocalized* 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, nonlinear – 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 will separate and take their original shape. Solitary waves which have this stability property are called solitons. The terminology stems from a combination of the word solitary and the suffix “on” which is used to signify a particle (think of the proton, electron, neutron, etc. ). We shall discuss a little later the sense in which a soliton is like a particle. Solitary waves and solitons have become very important in a variety of physical setting, for example: hydrodynamics, nonlinear optics, plasmas, meteorology, and elementary particle physics, to name a few. Our goal in this chapter is to give a very brief — and relatively superficial — introduction to solitonic solutions of nonlinear wave equations.

Taylor’s Theorem and Taylor Series (Appendix A)
Charles G. Torre
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.

Vector Spaces (Appendix B)
Charles G. Torre
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.

Foundations of Wave Phenomena: Complete Version
Charles G. Torre
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