Phase Pendulum

The Goldilocks problem

Notice that there is one pendulum in front of another.

When you grab the front pendulum and move it right and left, it moves the suspension point of the second pendulum right and left. You can use the front pendulum to drive the back pendulum side to side.

Hold the front pendulum and pull the back pendulum to one side, and release it, notice the frequency with which the back pendulum goes right and left, this is the fundamental resonant frequency, f0, of the back pendulum.

Move the front pendulum right and left much more rapidly than this fundamental frequency, notice that the back pendulum moves left and right with a small amplitude. When you move right, it moves left. We say the two pendulums are out of phase, or quantitatively, 180 degrees out of phase.

Move the front pendulum right and left much more slowly than this fundamental frequency, notice that the back pendulum moves right and left with a small amplitude. When you move left, it moves left. We say the two pendulums are in phase, or quantitatively, they have a phase difference of 0 degrees.

Move the front pendulum right and left as closely as possible to the fundamental frequency, notice that the back pendulum moves right and left with an amplitude which grows larger and larger with time. Eventually the pendulum will smack into the maximum amplitude stops. When you reach maximum right position, the pendulum is racing through the center. You are driving the pendulum at its resonant frequency and it is quantitatively 90 degrees out of phase.

As you move away from resonance the amplitude of the driven pendulum decreases.

If you drive it for a while at a constant frequency, the back pendulum will move back and forth at the same frequency you do.

Now you see why this is called the Goldilocks problem, too big or too small a driving frequency creates small amplitude motions, but just right, i.e. at resonance, results in a large motion. this is the essence of resonant amplification.

Why is the sky blue?

Air molecules (this is true for both nitrogen N2 and oxygen O2) have resonant frequencies of their outermost electrons which are in the ultraviolet frequency range. Pull the electron cloud of one of these molecules to the side and release it and it will oscilate back and forth at UV frequencies. As it oscillates it will emit UV radiation.

If you hit the outermost electron of an air molecule with red light, the red light will push back and forth on the electron at the frequency of red light, this frequency is below the resonant frequency of ultraviolet light and so the electron cloud will oscillate at the red frequency at a small amplitude. So, only a small amount of red light energy will be scattered.

If you hit the outermost electron of an air molecule with blue light, the blue light will push back and forth on the electron at the frequency of blue light, this frequency is below the resonant frequency of ultraviolet light but closer to it in frequency than the red light, and so the electron cloud will oscillate at the blue frequency at a larger amplitude than it did for red light. So, a larger amount of blue light energy will be scattered. Thus when white light hits air molecules more blue light is scattered to the side than red light and the sky is blue. (The light you see in the sky is sunlight that is scattered by air molecules into your eye.)

At resonance, the ultraviolet light causes such a severe motion of the electrons that the light is absorbed by the molecule. This region of the spectrum is known as the "vacuum" ultraviolet because it can only propagate in a vacuum, it is absorbed by air. It is also called ultraviolet C, ultraviolet with a wavelength shorter than 300 nm or a frequency over 1015 hertz.

What is the cause of refraction?

When light goes from vacuum into glass, the speed of propagation of the light is decreased. This slowing will cause the path of the light to bend at the surface. This phenomenon is known as refraction. Using the wave model of light from circa 1870 together with a more modern knowledge of atoms results in the following view of the cause of refraction. The atoms in the glass have a resonance in the ultraviolet. When the light wave strikes an atom it sets the electron cloud into oscillation, and loses some energy to the oscilation of the electron cloud. The oscillating electron cloud re-emits the light with a delay. Since we are driving the resonant electron cloud oscilation below resonance there is a phase delay between the driving light and the light emitted by the atom. If we look at the crest of a light wave as it enters the glass, every time the wave encounters an atom the crest is delayed a little bit. The continuous delays produce the apparent slowing of the light. Since blue light is closer to the resonant ultraviolet than red light it is slowed more at each atom, it's phase difference is greater, and so the index of refraction is a function of frequency and therefore wavelength. Some materials like iodine have a resonance in the infra red. For theese materials red light is closer to resonance than blue light and so red is slowed and refracted more than blue. This is called anomalous dispersion.

This dependence of the speed of light on wavelength is how prisms split light into its colors and how water drops create rainbows.

Math Root

Far from resonance the amount of incident light power scattered, P, is proportional to the fourth power of the frequency, f.

P a f4

Since blue light has about twice the frequency of red light, the power of the scattered blue light is 24 or 16 times grater than the power of the scattered red light. This type of scattering is called Rayleigh scattering.