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.

Origins of Newton's laws of motion

History of mechanical clocks with animations
Understanding a mechanical clock with animations
includes pendulum, balance wheel, and quartz clocks

## Saturday, July 9, 2011

### Derivation of constant speed of light from 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.

(6.1)

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.

Summary

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).