Their applications, physics, and math. -- Peter Ceperley
There are all sorts of resonances around us, in the world, in our culture, and in our technology. A tidal resonance causes the 55 foot tides in the Bay of Fundy. Mechanical and acoustical resonances and their control are at the center of practically every musical instrument that ever existed. Even our voices and speech are based on controlling the resonances in our throat and mouth. Technology is also a heavy user of resonance. All clocks, radios, televisions, and gps navigating systems use electronic resonators at their very core. Doctors use magnetic resonance imaging or MRI to sense the resonances in atomic nuclei to map the insides of their patients. In spite of the great diversity of resonators, they all share many common properties. In this blog, we will delve into their various aspects. It is hoped that this will serve both the students and professionals who would like to understand more about resonators. I hope all will enjoy the animations.
Fig. 18.1. Transformed Maxwell Equations as derived in the previous two chapters.
Unfortunately Maxwell's equations transformed as mixes of Maxwell's equations in the prime reference frame. We now need to separate the mixes. In order make the equations more manageable, we will use the abbreviations shown below in Fig. 18.1a. Note that each of the newly defined quantities are the left sides of the wished for Maxwell equations in the primed frame. We hope to show that the above equations (in Fig. 18.1 above) imply that all these quantities listed in Fig. 18.1a are zero.
Definition of m1, equal to the left side of the first transformed Maxwell equation.
Definition of the vector m2, equal to the left side of the second Maxwell equation (a vector equation).
Definition of m3, equal to the left side of the third Maxwell equation.
Definition of the vector m4, equal to the left side of the fourth Maxwell equation (a vector equation).
Fig. 18.1a. Definitions of variables used to separate out and isolate the various Maxwell equations.
Using the definitions in Fig. 18.1a, we can be expressed the equations in Fig. 18.1 as:
The first Maxwell equation as transformed written in terms of m1 and m4x.
The second Maxwell equation (a vector equation) as transformed and written in terms of the variables m2 and m3.
The third Maxwell equation as transformed and written in terms of the variables m3 and the x-component of m2.
The fourth Maxwell equation (a vector equation) as transformed and written in terms of the variables m1 and m4.
Fig. 18.2. Maxwell's equations as transformed by the above work, using the Lorentz transforms and written in terms of the variables defined in Fig. 18.1a .
We need to solve the above equations for the variables m1, m2x, m2y, etc:
Inspecting the equations, we see that four of the variables are equal to zero, i.e. (C2y), (C2z), (C4y), and (C4z). Thus ⇒
m2y = m2z = m4y = m4z = 0
Multiplying (C1) by V and subtracting it from (C4x) yields
m4x − V2·m4x = (1 − V2)m4x = 0 which implies that ⇒
m4x = 0 .
Substituting m4x = 0 into (C1) yields
m1 = 0 .
Similarly, we can multiply (C2x) by V and add this to (C3) to yield: m3 − V2m3 = (1 − V2)m3 = 0 which implies that ⇒
m3 = 0.
Inserting m3 = 0 into (C2x) gives ⇒
m2x = 0 .
Fig. 18.3. Solving for m1, m2x, m2y, etc.
In summary, all eight of the the variables m1, m2x, m2y, ... , m4z are zero. Using the definitions of these variables as listed above in Fig. 18.1a gives:
Fig. 18.4. Final result of the above work to transform Maxwell's equations using the Lorentz transforms. This shows that in the moving reference frame the laws governing electric and magnetic fields appear to be the same as they were in the stationary reference frame.
Thus, we have successfully shown that the Lorentz transforms when operating on Maxwell's equations, produce the same equations in the new reference frame.
Most text books would do this derivation much more efficiently using more advanced methods, briefly mentioned in Chapter 14. We have presented here this vector algebra derivation as an alternative, in part to illustrate that most physics can be derived in a number of ways and we can learn from each way.
Waves, Berkeley Physics Course - vol. 3, Frank S. Crawford, Jr. McGraw-Hill 1965. This book is suitable for an add-on to an introductory course on college or university physics. It discusses all sorts of aspects of waves and has a multitude of home experiments. One could probably make a great science fair project from one of them. As to its math level, it mostly uses algebra, with some calculus in the mix.
Physics of waves, by Elmore and Heald, originally published by McGraw-Hill in 1969, but currently published by Dover. This book covers many different wave systems, such as waves on a string, on a membrane, in solids, in fluids, on a liquid surface, and electromagnetic waves. It also covers the many aspects of waves. It has an excellent chapter on diffraction.
The Feynman lectures on physics, Feynman, Leighton, and Sands, Addison-Wesley 1963. Three volumes. These cover many aspects of physics. They are perhaps best suited for someone who has made it through an introductory sequence in college or university physics, and wants to read about the subject from a more sophisticated point of view. They are not particularly math intensive, more just into discussing concepts with some math as required. These are books you read to understand a physicist's mind. Perhaps 10% to 20% of the chapters are about waves and resonances.
Electromagnetic books that I use:
Engineering Electromagnetics, Hayt (with Buck on more recent editions), McGraw-Hill. An easy to read, compact junior-level text for electrical engineering students.
Fields and waves in communication electronics, Ramo, Whinnery, Van Duzer, Wiley. A upper level/graduate level text for electrical engineering student. Covers practically every aspect of applied electromagnetic fields in some depth. Is not a book to sit down and read for philosophy, but rather to look up the rational behind certain devices or design methods.