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.

For a list of all topics discussed, scroll down to the very bottom of the blog, or click here.

Origins of Newton's laws of motion

Non-mathematical introduction to relativity

Three types of waves: traveling waves, standing waves and rotating waves new

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

Water waves, Fourier analysis

Monday, October 29, 2007

The technology of oscillations and waves demands new mathematical methods.

Thomas Edison George Westinghouse

The true utility of Euler's work was not evident until the mid to late 1800's. At this time George Westinghouse, Charles Steinmetz, and others needed a way to conveniently calculate quantities associated with the newly invented AC power. Previously, people had only used DC power, and found attempts to implement AC to have unpredictable and often deadly results. Indeed, Thomas Edison, the famous inventor of the light bulb, called AC power the work of the devil, deeming it useful only for killing people.

Today, by using complex numbers, electrical engineers and physicists can easily calculate the behavior of electrical oscillation, AC power, and communication signals and circuits. A major fraction of modern technology, that which uses waves and oscillations, is made possible by Euler's invention.

design of modern electronics depend on complex analysis of oscillations and waves

To understand the utility of Euler's complex numbers for waves and oscillations, let's consider a typical "sine" or "cosine" wave. This may be the mathematical representation of an oscillation, a vibration, a resonance, or a wave. Typical applications include acoustical waves, AC power systems, laser and fiber optics communication systems, radio and television components, and cell phones. Below are shown some oscillations that can be mathematically described by cosine waves.

Animation of meter reading voltage of a typical household power socket. Mouse over the image to see the action. You need a flash player installed on your computer to see this animation.

Notice the pointer swings wildly back and forth from positive voltage to negative voltage and back again in cycles. This is because our society uses "AC power" where AC stands for "alternating current". Use of AC power allows us to use transformers to jack up the voltage to levels of hundreds of kilovolts for efficient delivery of power from power plants hundreds of miles away, and to return it again to relatively safe levels for use in the household. Transformers do not work with simple, steady, "DC" power. Without use of AC power and transformers, power plants would have to be distributed though out large cities and the use of distant power sources such as from dams, water falls, or wind turbines would be impractical, because most of their energy would be lost in bringing the power to the cities.

For the purpose of animation, the rate of oscillation has been slowed way down. Real AC power reverses direction 120 times a second, too fast for the human eye to follow, and requires a special "AC" setting on a voltmeter. AC meters read an "RMS" voltage which is similar to the "amplitude" that we will discuss below.

Development of AC power required the simultaneous development of the mathematical method that we will be describing in this lesson. Because of its complexity (which was eventually overcome by these mathematical methods) Thomas Edison never did accept accept AC power and considered it the work of the devil.

Electronic audio signals, such as those going through your stereo, walkman, iPod, and cell phone oscillate in much the same way, although their rate of oscillation is much greater and various with the pitch of the sound being generated. They reverse direction 500 to 10,000 times per second. TV,radio, and cell phone transmission signal also oscillate in a similar fashion, at a rate of 1 million to 5 billion times per second. Laser signals (such as those that carry most of the internet traffic in optical fibers) reverse at a rate of several trillion (2 to 6 × 10 12) times per second. All these types of signals can can be analyzed by these methods.

voltage in ac power as a function of time, cosine function

Usually such a wave, or oscillating voltage fluctuates up and down in a very regular pattern, just the same way as the cosine function does, except here the function is plotted versus time instead of angle.. Electrical engineers and physicists use a measuring device called an oscilloscope to make graphs of rapidly oscillating waveforms. Oscilloscopes are amazingly fast and can make graphs of oscillations that reverse up to several billion times per second, as well as those slower than that.

The mathematical formula for such oscillation is (written for the voltage for the sake of definiteness):

equation for the voltage as a function of time,

where A is the "amplitude" (how tall the oscillations are), ω is the "angular frequency" (a measure of how fast the voltage oscillates), and ϕ is the "phase shift" (a measure of how much the peaks are shift over from the starting y-axis. You can alternately use sine functions to express the same wave, or a mix of sines and cosines. We should point out that the argument of the cosine function, the ωt + ϕ in the above equation, is in the angular units of radians as discussed in the first animation. For a fairly detailed discussion of phase click here.

We can use this mathematical representation for all sorts of waves, oscillations, vibrations, and resonances. We sometimes call this (and the yet discussed complex equation for waves) a "wavefunction". Technically, true waves, as opposed to oscillations and vibrations, require a slight modification of the above equation, but we won't get into that now.

The problem comes when you want to add two waves together, or calculate the effect on the wave of passing through a particular wire or component. These types of mathematical manipulations are typical of what is needed in calculation to design wave related technology, such as AC power components or systems. Doing the necessary calculations with sines and cosines is quite tedious. We shall next demonstrate this to make our point.

To understand the problem involved, consider the two waves shown below

two oscillating wave forms to be added together.

The sum of the red and blue wavefunctions added together is the green wavefunction shown at the right. It is not very obvious to the average person that this is indeed the sum.

sum waveform Unless you do this a lot, it is difficult to simply glance at two wavefunctions and know what their sum will be. Wavefunctions, when added together can do strange things. They can "constructively interfere", "destructively interfere", or do a combination to the two.

Ok, so it is complicated, but how do we sum the two waves mathematically?

Summing two waves using trigonometric functions.

Mathematically we can expression their addition as:

equations for summing two cosine waveforms.

Experimentally, we know that the sum of two waves of equal frequency always results in a third wave of the same frequency, which we can generally express as:

equation for sum waveform.

Now the puzzle is: how to solve for the resulting amplitude, A3, and phase, ϕ3 ,in terms of the known quantities A1, ϕ1, A2, and ϕ2.

This can be a very difficult problem, but I will lead through by perhaps the fastest way (fastest without use of complex numbers). There are many slower methods and ways to get hopelessly lost.

We start by applying the trigonometric relationship:

trig identity for cosine of a sum of two angles.

This relationship is not something the average person would be expected to know, however the ancient Greeks (Ptolemy) did discover it. Applying this relationship to the two above formulas for v3(t), we get:

equation 1, reducing sum waveforms

equation 2, reducing sum waveform.

The first equation can be factored as:

equation, factored sum waveform

The time dependent parts of the equations are the cos ωt and sin ωt factors. In order for the upper equation to equal the lower one at all times, the coefficient (or multiplier) of the cos ωt in the first equation must equal the cos ωt coefficient in the second equation. Likewise for the sin ωt factors. Thus, we have the two equations:

equation, cosine factors

equation, sine factors.

We need to solve these two equations for the two unknowns A3 and ϕ3 . We can eliminate ϕ3 and solve for A3 by squaring both equations and then adding them, remembering that sin2ϕ + cos2ϕ = 1. This gives us:

equation for amplitude squared of sum waveform.

Taking the square root yields:

equation for amplitude of sum waveform

Similarly, we can eliminate A3 from the two equations and solve for ϕ3 by dividing one equation by the other:

equation for tangent of sum waveform

equation for phase angle of sum waveform.

illustration, frazzled Phew! That was a lot of work! At this point, you should understand the need for a faster method of adding wavefunctions together. While the above solution is correct, it is neither fast nor transparent. For something as important as the technology of waves, we need a faster, more transparent method to add two waves. This better method will be with "the complex representation". Before we get into that, we shall discuss an intermediate representation that takes us half way there.

NEXT: Phasors to the rescue.

© P. Ceperley 2007

Good references on WAVES Good general references on resonators, waves, and fields
Scroll down farther for a list of the various related topics covered in postings on this blog.