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- Wave equation (4)
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Articles 1 - 30 of 35
Full-Text Articles in Physics
03 How To Find Normal Modes, Charles G. Torre
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
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
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
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
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
08 Fourier Analysis, Charles G. Torre
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
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
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
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
Problem Set 7, Charles G. Torre
Vector Spaces (Appendix B), Charles G. Torre
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
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
Problem Set 10, Charles G. Torre
Problem Set 8, Charles G. Torre
01 Harmonic Oscillations, Charles G. Torre
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
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
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
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
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
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
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
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
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
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
Problem Set 6, Charles G. Torre
References And Suggestions For Further Reading (Appendix C), Charles G. Torre
References And Suggestions For Further Reading (Appendix C), Charles G. Torre
Foundations of Wave Phenomena
References and Suggestions for Further Reading (Appendix C)