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.
8. Derivation of the Lorentz transforms starting with length contraction and time dilation
We begin by explaining the Galilean relativity motion transforms and then go on to derive the Lorentz transforms, both shown in Fig. 8.1 below. Both sets of transforms assume that the unprimed reference frame (with coordinates x, y, z, and t) is stationary and the primed frame (coordinates x', y', z', and t') is moving in the x-direction at constant velocity V.
Fig. 8.1. Table of Galilean and Lorentz transforms.
The length contraction and time dilation formulas are:
, (8.1) and
Δt = γ Δt' , (8.2)
where the relativistic factor γ is given by:
The x transform:
The first Galilean transform is easy to understand. It says that a stationary point in the stationary reference frame having x-coordinate value of x will appear to be sliding to the left from the point of view of someone in the moving frame. In that reference frame, the point will appear to shift an amount −Vt, i.e. its x' coordinate value will be given by the initial x value plus this shifting amount:
x' = x − Vt (8.4) .
In the relativistic case we need to account for the length contraction in the x-direction. That is, the x' scale will be shrunk and the same value in the x coordinates will translate to larger x' values because of this shrinking. The relativistic factor
is needed here to multiply the x' coordinate in (8.4) and complete the transform:
x' = γ (x − Vt) (8.6) .
The y and z transforms:
The y and z transformations are really not transformations at all, because there is no relative motion in these directions and also no length contraction in these directions. So in both the Galilean and relativistic cases:
y' = y and z' = z (8.7) .
The time transform:
In the Galilean case, the time transform is trivial, i.e. there is no transform and t' = t.
On the other hand, in the relativistic case, deriving the time transformation takes some work. The first term in the Lorentz time transform is easy: it is just the time dilation factor γ applied to the original time t.
t' = γt (8.8) (incomplete!).
So what about the extra −Vx/c2 term in the last Lorentz transform equation in Fig. 8.1? It might be called the synchronization term. It insures that time at the various points in the moving frame will appear to be synchronized to a person in that frame, as you might expect.
We would expect our time to be synchronization, that time did not seem to vary with the coordinate x . However we saw in the previous chapter that when we change reference frame, previously simultaneous events are no longer simultaneous, if they have different x values.
In the previous chapter, we found that two events that are simultaneous in the stationary frame will have a time difference between them of Δt = γΔxV/c2 (see Equation (7.11) ). If we wish to correct for this time difference we need to subtract it from t in Equation (8.8):
t' = γ (t − xV/c2) . (8.9)
Equation (8.9) is the time part of the Lorentz transformations. Without this last term, a person on a moving frame would sense that clocks located in different x-positions in his frame would not be synchronized, assuming they were synchronized the stationary frame.
Simultaneity in high speed reference frames:
Because of the last term in (8.15), simultaneity is not longer a simple issue at relativistic speeds, i.e. at speeds close to that of light. At low speeds, if two events occur at the same time, i.e. simultaneously, then they occur at the same time in all moving reference frames, even if the two events are at two different locations. Thus if lightning strikes two different trees at the same time as witnessed by a stationary observer, then an observer flying at low speeds overhead will also see the two strikes occurring at the same time assuming the observers are about the same distance from the two strikes. On the other hand, if an airborn observer is traveling at relativistic speeds, even if he is equidistant from the two strikes (that are simultaneous in a stationary frame), the two strikes will not appear occur at the same time to this flying observer.
The last term in (8.15), i.e. the synchronization term, means that for high speed motion, the time in a high speed moving reference frame depends of both the x position and on the velocity of the frame, V. Thus, two events occurring at the same in the stationary frame, so that t1 = t2, will not have the same time in the moving frame if they have different x coordinates which make the last terms in (8.15) differ and in turn make t1' ≠ t2'. In general, for relativistic speeds, simultaneity of two events that have different x coordinates, only holds in a particular reference 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.