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.
We need to use inverse Lorentz transforms in several of the derivations in the coming chapters, so we derive these here. The Lorentz transforms from the previous chapter are:
These transforms give the primed or moving coordinates as functions of the unprimed or stationary coordinates. We will invert these to give the stationary coordinates as functions of the moving coordinates.
Inverting y and z transforms:
The y and z coordinates, (4.1b) and (4.1c), are trivial:
y' = y inverts to give y = y' and
z' = z inverts to give z = z' .
Inverting x and t transforms:
The transforms for x and t are intertwined and must be handled together. We repeat the original x and t transforms:
x' = γ (x − Vt)   (4.2a)
t' = γ (t − Vx/c2)   (4.2b)
We solve (4.2a) for x:
x = x'/γ + Vt (4.3)
and substitute this into (4.2b):
Multiplying (4.4) by γ yields:
This can be solved for t:
Substituting (4.5) into (4.3) yields:
The factor in parenthesis in (4.6) can be reduced as follows:
Using this in (4.6) gives:
Inverse Lorentz transforms
x' = γ(x − V t) (4.8a)
x = γ(x' + V t') (4.9a)
y' = y(4.8b)
y = y'(4.9b)
z' = z(4.8c)
z = z'(4.9c)
Equations (4.9a) through (4.9d) are the inverse Lorentz transforms. They give the stationary (unprimed) x and t coordinates in terms of the moving coordinates (the primed ones). They are also interesting because except for the minus signs in front of the second terms in each equation, they are identical to the original Lorentz transforms, Equations (4.8a) through (4.8d). They can be had from the original transforms by simply substituting −V in place of V. It means that to the moving reference frame, except for the stationary frame appearing to be moving in the opposite direction, it has the same length contraction and time dilation that people in the stationary frame see when looking at the moving frame.
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.