Resonant Frequencies
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Resonance is Everywhere (Ubiquitous)
I am hard-pressed to think of a system in nature that does not have some sort of resonance associated with it. Light itself is a resonance phenomenon, atoms and molecules both have many different resonances associated with them, large structures made of the same things (atoms and molecules) also resonate. Air resonates, entire oceans resonate under tidal forces, planetary orbits can resonate with others, and stars do the same. Models show that galaxies have tidal resonances. Galactic resonances due to outward radiation pressure and dark energy being in opposition with gravitation from ordinary and dark matter also are expected to exist.
Why things resonate in a mechanical way comes down to basically this: Any system that is stable and persistently exists in our universe does so in spite of its environment. The environment constantly pushes and pulls on the system and the system resists collapse due to these forces. Matter in general is elastic - even materials that you think of as brittle. Like a kid on a swing which always tends to come back to center (equilibrium) upon being pushed, materials have the same tendencies. Bend a stick and it straightens itself. Of course there is a breaking point for just about everything, but that is only at energies beyond which resonance no longer occurs and beyond which the system will be destroyed. Subsequently the parts or pieces of the previous system will, however, have new resonances of their own.
Maybe I should mention briefly the utility of resonating something to the point of destruction. In the same way that a singer can actually shatter a wine glass with their voice resonating at the frequency of the wine glass, when companies want to drill deep into the earth in search of natural resources, they exploit resonances of the rock layers in order to break them down. This saves enormous amounts of money in drill bits. This method is often called "sonic drilling", and the drill head itself is often also the source of vibration that induces the resonance of the ground.
A good way to think of resonance is in terms of waves as we have done in earlier chapters, but to tie those waves and associated phasors to energy is also a good idea. I mentioned briefly in the last chapter that resonance is a means of effectively transferring energy to a system. This is true for electrical circuits, light waves, sound waves, kids on swings at the park, etc. There will always be some frequencies of stimulus (or driving force) which more effectively transfer energy to a system than others.
Mass-Spring System
Recall from our studies of mechanics that a spring attached to a fixed support on one end with a mass on the other end will vibrate if pushed away from equilibrium. In fact, we will find out in our studies of quantum mechanics that it can't actually ever stop oscillating, but that's a discussion for another day.
Using Newton's second law and Hooke's law for the spring force, we can write . By writing acceleration as the second derivative of position, we can transform this into a differential equation:
The solution to this equation is a linear combination of sine waves and cosine waves of arbitrary amplitude. Given initial conditions we can narrow down a more specific solution. One solution with original amplitude A and zero initial velocity is where This term is the resonant frequency of the system - obviously described as angular frequency and not ordinary frequency f. The idea is that the system naturally tries to oscillate at this frequency. If you try to shake it at some other frequency it will move, but will not acquire a very large amplitude, which is another way of saying that it won't acquire much energy from the shake. The better the match between the shake and the resonant frequency, the more effectively energy will be transferred. Let's look at the intensity of the vibration in terms of these parameters.
Suppose a system resonates at a frequency while being driven by something trying to induce, or drive, a vibration in the system at a different frequency . There is always some damping or dissipation of energy of the vibration to some other form in natural systems. This damping will be denoted by . The equation plotted below shows intensity on the y-axis and the driving frequency on the x-axis. The message of the plot is simple: Try to give a system any vibrational intensity by shaking it at a frequency too far away from its natural resonant frequency and it won't work. It is easier for heavily damped systems where k is large (as you can see if you play with the slider), but still the best is to match the resonant frequency. It's like a child on a swing. You push at the right time and they go high, at the wrong times (you'd need longer arms to do this) and it'll basically mess up any amplitude of swing they have already.
A plot of the response (amplitude or intensity) versus driving frequency is called a frequency response curve or frequency response spectrum. The idea is that the curve indicates how a system responds to energy being input to the system at a given frequency. If you recall that resonance is the frequency-dependent ability of a system to absorb energy, then these response curves are literally a picture of the resonances of a system.
The one below is what you may measure for a child on a swing. Notice that there is only one frequency that produces any significant amplitude. In lab you will plot a frequency response curve for a string under tension. In that case you will find several frequencies at which resonance will occur.