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.
6. Derivation of the constant speed of light from the Lorentz transforms
Velocity of light in the x direction in the moving frame:
We consider a light pulse traveling in the x direction, crossing the point x1 then the point x2. The distance traveled is related to the time it takes by:
x2 − x1 = c (t2 − t1) .
We use this and the Lorentz transforms (4.8) to calculate the velocity of light in the primed or moving reference frame.
Equation (6.1) shows that in the x direction, the velocity of light is the same value in the moving frame as in the stationary frame, i.e. c' = c .
Velocity of light in the y or z direction in the moving frame:
We next derive a similar result in a direction perpendicular to the relative motion. We will use the y direction for convenience, but we could equally well use any perpendicular direction.
We start by noting that to get light to appear to go vertically in the moving frame, it must be angled in the stationary frame as illustrated in the sketch at the right. This is similar to rain in a windless day coming straight down, but appearing to be coming down at an angle directed towards you when viewed from a moving automobile or bicycle.
As noted on the sketch, we immediately have two relationships:
Δx = VΔt and
diagonal = cΔt .
Using Pythagoras's theorem, we solve for the remaining side:
This can be solved for Δt: Δt = γb/c .
Now to calculate the perpendicular velocity of light in the moving (primed frame):
Note that we also used the fact that the y coordinate is unchanged during the Lorentz transformation, making Δy' = Δy = b .
Thus, c' = c . As it was in the x direction, the velocity of light in the y direction is not changed by relative motion.
The velocity of light is unchanged by Lorentz transformations in both the parallel and perpendicular directions. We will address this question again for the arbitrary direction case after we have derived transformations for velocity (in Chapter 9).
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.