N- Coupled Harmonic Oscillators
In the last article, I introduced the concept of a “coupled” oscillator, and how its vibrations can be considered as a form of energy transmission between two masses. Coupled oscillators are, indeed, one of the most beautiful concepts in all of physics, perhaps because there is more than one of them. A single oscillator might be easy to model, but two, or three, or even more oscillators are much more rigorous, and lead to some stunning truths about the nature of wave phenomenon. This is, obviously, a continuation of the previous article, in which I left off with a note about normal modes. In a coupled two-oscillator system, there are two possible normal modes which can exist — an in-phase mode corresponding to angular frequency ω1 and an out-phase mode corresponding to angular frequency ω2. We also found out the exact values of these normal mode resonant frequencies.
So, it is completely natural to wonder, what if we had three masses? Well, in this article we will start off with the physics of a coupled three-oscillator system, in the same vein as the last two-oscillator system. And, after that, we shall go on to experience the wondrous beauty of physics and complex harmonic phenomenon.
THE THREE OSCILLATOR SYSTEM
Again, we shall start off our exploration by detailing the basic setup. Investigations into physics are almost necessarily thought experiments, and it is always a good idea to see what we’re dealing with. So, what are we dealing with here? Well, the setup is the same as before. We have three masses of equal mass m, connected to each other by springs of stiffness k. Further, the two edge masses are linked to a wall by a spring, giving us four springs in total. Let the natural frequency of each spring be ω. The positions of the masses are x1, x2, andx3. We shall consider the free oscillation of the masses in question when set to some initial conditions.
To analyse this system, we need an equation of motion. We can combine the EoMs of all three oscillations into a single matrix differential equation, as shown below.
To solve this, of course, we need to know two things — the eigenvalues and eigenvectors of the matrix. They are as follows
The solutions to equation (1) are, similar to a two-oscillator system, as follows:
As you can see below, the graphs of x1(t), x2(t) and x3(t). Just like the two-oscillator system, energy is transferred between each mass, and three normal mode frequencies exist. They are:
As shown before, there are three natural frequencies, or normal mode frequencies, available to this system. Indeed, similar to the two-oscillator system, in which we had the following modes of vibration as represented as eigenvectors:
These two, of course, represent the in and out of phase solutions respectively. When a two-oscillator system is forcibly oscillated at its eigenfrequencies, it undergoes simple harmonic motion in the above modes of oscillation. The same can be said about a three-oscillator system. Its eigenmodes, in order of increasing frequency, are as follows:
So, what exactly are these eigenmodes telling us? Well, the first one M1 tells us that, at the lowest resonant frequency, the coupled oscillator system moves in-phase, i.e., all masses move in the same direction (albeit with varying amplitudes). The second mode, M2, tells us that when the system is oscillated at ω2, the first and third oscillators move in antiparallel directions, while the second one doesn’t move at all. The third mode, M3, tells us that, when oscillated with frequency ω3, each pendulum is out of phase with the one adjacent to it.
Now that we have looked at the eigenmodes of a three-oscillator system, we will be spending the rest of the article analyzing only the eigenmodes of n-oscillator systems.
NORMAL MODES OF N-OSCILLATOR SYSTEMS
Let us now consider not just one, but n coupled oscillators attached together in a finite line. How can we model the motion of this oscillator? Well, we can do the same exercise as before. If the positions of the masses are x1…xn, then the equation of motion for the system becomes
As we see, there are, obviously, n separate equations of motion, one for each oscillator. As such, we get an n x n matrix that has n separate eigenvectors and eigenvalues and finding out each one and summing them all up to find a homogenous solution is simple in principle, but not something you want to sit down and do. So, what do we do? We only analyze motion in the eigenmodes of the system, and let a computer calculate the eigenvectors for us. Below are the eigenmodes for a four-oscillator system, by increasing frequency.
Now, here are the eigenmodes for n=5. Is there a pattern here?
Let us look at the first eigenmode for n=5. At this least frequency, all masses are in phase with each other. Indeed, if we plot the displacement of each mass as a function of n, the nth oscillator, we get a symmetric curve with one crest that’s always positive.
In the second mode, we see that the first two masses oscillate in phase, and so do the last two, but out of phase with the first two. Indeed, if we plot this on a chart, we see that we get a distribution with two extreme values — one peak and one trough. For the third mode, as you might have recognized already, we will get threeextremes — two peaks and one trough. Thus, if we are looking at the nth harmonic, we realize that there will be n extreme values. That is the pattern here, and the charts will, as you might have guessed, oscillate up and down.
We have just figured out a pattern here. It may not be clear right now, as we are only working with discrete sets of oscillators, but this pattern holds the foundation of not just classical wave mechanics but offers us a peeping hole into quantum mechanics as well. Indeed, after this article and the next, we shall realize what the quantum in quantum mechanics truly means.
FROM DISCRETE TO CONTINUOUS
After looking at the eigenvector data before, we concluded that not only do the masses oscillate for various resonant frequencies, but that the amplitudes themselves oscillate depending on which mass one is looking at. It is natural to extend this observation to a system of infinitely many coupled oscillators, which gives us infinitely many normal modes as well. Now, here we get to a point about data: if I have a set of data based on discrete independent variables, then the more of the independent variables I take, the data essentially becomes continuous. As a side note, this is the fundamental concept behind the Fourier transform, but we shall not look at it here (it is, however, pertinent in looking at such systems at a much deeper level).
Our first observation? If we are looking at the nth mode, or harmonic as we shall now call it, we will have exactly n extreme values, and if our system is continuous, the displacement from mean of each oscillator as a function of distance along the line will be a sinusoidal graph — peaks and troughs, after all. Now, we get to beauty that lies in this investigation — if we have frozen the motion of this continuous oscillator in time, we get a sinusoid as a function of position x. This sinusoid contains an integer number of half-wavelengths. Further, this sinusoid isn’t stationary, obviously. The individual displacements y vary with time as well, and we know for a fact that those displacements will, indeed, be sinusoidal. Therefore, we get a wave y(x,t) that varies with both space and time, but doesn’t leave the confines of the oscillator.
Have we seen this particular concept before? We have. We definitely have. This is a standing wave, and we have derived the concept of waves in infinite oscillator systems, not in some hand-wavy manner, but through careful observation of simple, discrete phenomenon that we have made continuous.
And that is one of the most beautiful things about physics — modelling phenomena from ground up. We started out with just one oscillator, then two, then three, and we have not just modelled a pretty common phenomena, but we have rationalized it through a comparison with another phenomena. And that is where the beauty lies — you see one thing in nature, and you say to yourself, “Hey, this reminds of something else!” The next logical step is to always model that system and see what happens if I try to make it look more and more like the first system you looked at. Indeed, you might have noticed that I haven’t gone into the mathematics of a standing wave yet — that will be another article. Why? Because not only will we be looking at waves, but we will also be considering the phenomenon of quasiparticles — yes, particles from waves. Does it remind you of another branch of physics?
If it does, good for you. But it shouldn’t. Why? Because we are still firmly within classical mechanics. We aren’t at tiny scales at all — but that will be explored in the next article.