Chapter 30: Inductance; and Electromagnetic Oscillations Applications

"The Swing" by Jean-Honore Fragonard hangs in the National Gallery of Art, Washington DC

Resonance

The term "resonance" appears over and over in physics. In your text, resonance has been mentioned (so far) in sections 14-8 (mechanical oscillations), 15-9 (wave motion), and 16-4 (sound). In this chapter and the next, it will come up again in connection with electrical oscillations. In the future, resonance will be a part of our discussions of light. You may also have heard the term used in connection with various technologies such as MRI, which stands for "Magnetic Resonance Imaging." At first, it is tempting to think that these phenomena are unrelated, but this is not the case. In this essay, we will explore resonance in general terms, drawing examples from several of the categories mentioned above.

One place to start is with the meaning of the word itself. Resonance is derived from the latin roots re- (again) and sonare (to make a sound). A resonance is literally an echo. This is actually a better description than it may seem at first. Resonance occurs when a system echoes energy that is put into it from an outside source.

Waimea Canyon on Kauai, Hawaii. Photo credit: the National Oceanic and Atmospheric Administration, Commander John Bortniak, Photographer.

As an example, imagine that you shout into a canyon, an empty hallway, or any other place that has a good echo. A moment later you hear your own voice back, andf at regular times after that, you hear your voice again and again, getting progressively fainter. Now imagine that you shout once, wait the appropriate amount of time, then shout again at just the moment the first echo is due. To a person standing with you, the result would sound like an extra loud shout. Furthermore, after a second, equal, time lag, you would hear an extra loud echo of your "double shout." If your shouted a third time.... You get the idea.

Now, let's consider how this shouting is connected to masses on springs, pendula, and all of the other phenomena mentioned above. The most striking similarlity is the presence of something periodic. In the case of shouting, it is the periodic echo of our original sound. Mass-spring systems, the molecules in a sound wave, or the charge in an electrical circuit all change position in a periodic way. Another similarity is the periodic ways of adding energy to these systems. (For the molecules in a sound wave, periodic shouting is one of those ways.)

Now, we can see that two things are necessary for resonance. First, we need some sort of a system that can undergo periodic changes. Second, we need a method for adding energy to that system that can be "timed." That is, we need to be able to adjust the frequency of the energy input. This second requirement allows us to make sure that all of our energy inputs go in at the right frequency, so our system's echoes are used to good effect.

Obviously, a very important part of creating resonance is picking that special frequency that allows energy inputs to be added to one another in the system. This special frequency is called the "natural frequency" or, in some books, the "resonant frequency." For the simplest systems this was the only frequency that was discussed, so it may not have been obvious that other motions are possible.

Recall the example of a pendulum, which was discussed in Chapter 14. If the bob is pulled back and released, it will oscillate at its natural frequency, ₀=(g/l)^1/2, where g is the acceleration due to gravity, and l is length of the pendulum. However, it is possible to force the pendulum to oscillate at a different frequency. The figure at the left suggests one way. In this system, the hand moves back and forth with angular frequncy , where is not necessarily equal to ₀. In this case, the pendulum will oscillate with frequency , and with an amplitude that is dependent on the amplitude of the hand's motion.

To see how resonance in this system works, let's mentally simplify the driving force. Instead of a sinusoidal drive at frequency , let's think about a force that consists of sharp smacks to the right that occur at fixed intervals (much like our shouts at fixed intervals). Initially, the pendulum is at rest. Then, at t=0, it gets its first smack. It swings off to the right, then gravity starts to pull it back. If then next smack occurs while the bob is still moving to the left, then all it does is slow the bob down, it actually takes away some of the energy that the first smack gave. Clearly, this is no way to build up a large amplitude.

If the second smack occurs while the bob is moving to the right, this is better, but with the bob moving away from the blow it still doesn't get all of the energy that it could. (It "rolls with the punch" boxers and stunt actors do this all the time to avoid injury). What is even worse, with the second blow mistimed by a little, subsequent blows will be mistimed by even more. Eventually, they will be opposing the motion, slowing the bob down as in our first example. Again, these mistimed blows will never lead to a really large oscillation.

The ideal situation would be if the second blow came right when the bob has come as far to the left as it will go, when it is momentarily stationary. At this point, the force of the smack comes right when the bob can accept it all, and right when it is added to the maximum force to the right coming from the string. The analysis for the third smack, and all later smacks is the same. If the blows are all well timed, they each add energy to the system, and a large oscillation results. The conclusion is that it is best if the smacks are timed so that they have the same periodicity as the the bob's natural motion.

This helps to explain why energy is most easily transferred to the system at its resonant frequency. Once any energy has been transferred to the system it starts to move. Energy coming in at the natural frequency is always added to the system. At other frequencies, the energy can only be accepted for a fraction of the time, at other times, the system tends to dump energy back into the source, as when the bob is moving to the left while receiving a blow to the right.

This may seem confusing, but if you have ever pushed someone on a swing, you have experienced it yourself. The best way to get someone swinging is to push each time they have swung back and are stationary near you. If you try to push while they are still on the backswing, they get a jarring blow in the back, and you strain your wrist. Not much swinging gets accomplished.

Another obvious question is "Why should we care about resonance, how can it be used"? In most devices that make use of resonance, the idea is this. Energy is provided to the device in a "disorganized" way. Rather than energy arriving all at one frequency, it arrives as a mixture of many different frequencies. A resonant system is then used to "pick out" the energy at its resonant frequency.

A great example of this is a radio tuner. Your radio antenna is just sitting there exposed to the radio waves from all of the stations in your area. It responds to all of them equally. However, inside your radio there is a tuning circuit that has a resonant frequency given by ₀=1/(LC)^1/2 (see section 30-6). This circuit takes in the part of the radio energy arriving at your antenna that has the frequecy of some particular station, and rejects the energy from the other stations. Thus, you can listen to your music, without being bothered by the music that other people prefer.

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