In this video, I want to discuss the phenomenon of resonance and show you a simple model for it. What is resonance? We can imagine a simple instance of it. If you have a crystal wine glass and you tap it, you can hear pretty clear tone, a pretty clear frequency associated with that wine glass. That's the natural frequency of vibration. Now, if someone is going to sing exactly at that frequency, they can cause that wine glass to vibrate somewhat violently until the wine glass breaks. Let's look at a video of that. Now remember, wear safety glasses. Now that you've watched the video, you see what the phenomena of resonance is about. It's very difficult to model a three-dimensional wine glass, but it's easy to write down a one-dimensional model of resonance. Let me show you what the differential equation looks like. If we look at x double dot plus omega naught squared x equals f cosine omega t, the homogeneous part of this equation, x double dot plus omega naught squared x equals zero, is the wine glass by itself. It has a natural frequency omega naught. The f cosine omega t is modeling then the voice that the person is singing at a frequency omega. Resonance is going to occur when omega gets close to omega naught or in the limit that omega goes to omega naught. So, that will be our model. So, the idea here then is to solve this equation This is a second order linear constant coefficient equation that has an inhomogeneous term. So, we know how to solve that. We will solve this equation when omega naught and omega are at different frequencies and then we'll limit the solution as omega goes to omega naught. Let's begin with the homogeneous solution. So, the homogeneous solution, we can go through our usual procedure of finding it but it's actually quite simple in this case, which function when you take the second derivative, you get back the function itself and the negative of omega naught squared. So, we know that the second derivative of cosine is negative cosine and the second derivative of sine is negative sine. We can write down the solution here as some constant times cosine omega naught t. The second derivative will give us minus omega naught squared times the function back, plus another constant times sine omega naught t. That's the homogeneous solution. To that, we're supposed to add the particular solution. To find the particular solution, we have an ansatz. So, we can try for the particular solution. Typically, when there's a cosine omega t on the right-hand side, we would try a constant times cosine omega t plus a constant times sine t. In this case, it's a little bit easier because there's no first derivative on the left-hand side. So, the second derivative of cosine gives us cosine. X gives us cosine, so we don't need to introduce a sine. For our ansatz, we can just try a times cosine omega t. We can substitute into the differential equation. We get minus omega squared A times cosine of omega t plus omega naught squared A times cosine omega t equals f times cosine omega t. So, we can cancel the cosine omega t and we can solve for A. So, A then is equal to f over omega naught squared minus omega squared. Okay, putting together the homogeneous solution and the particular solution, we get the general solution for x n is equal to C_1 cosine omega naught t plus C_2 sine omega naught t plus the particular solution plus f over omega naught squared minus omega squared. That's our A times cosine omega t. That's our general solution. Okay. If we consider omega goes to omega naught, it's a funny expression here. We're not yet done. If omega goes to omega naught, we don't really know what to do with this denominator here. So, what we need to do is we need to satisfy the initial conditions. We have x of 0. What should be the appropriate initial conditions for a wine glass? We have the wine glass is just sitting there. So, no displacement, no velocity. We have x of 0 equals x dot of 0 equals 0, would be the reasonable initial conditions. So, we substitute in x of 0. We get C_1 and sine 0 is 0. Cosine 0 is 1. So, C_1 plus f over omega naught squared minus omega squared is supposed to be equal to zero. We try x dot of 0 equals 0. The cosine becomes a sine, that will be zero. The sine becomes a cosine. We have omega naught C_2. The cosine becomes a sine will be a zero. Omega naught C_2 equals 0. So, C_2 is equal to 0. C_1 is equal to minus f over omega naught squared minus omega squared. We can write down the solution here. Let me put it up here. We have x of t is equal to C_1 which is minus f over omega naught squared minus omega squared cosine omega naught t, plus f over omega naught squared minus omega squared cosine omega t. We can write this in a nicer way. We can factor out f over omega naught squared minus omega squared and then we have a cosine omega t minus cosine omega naught t. Okay. So, that's the solution to the differential equation when omega is different than omega naught. We are interested in resonance now when the singing voice matches the natural frequency of the oscillator. We'd like to know what happens when omega approaches omega naught. We have cosine omega t minus cosine omega naught t, that will become cosine omega naught t minus itself, would be zero. We have omega naught squared minus omega squared. Omega goes to omega naught. That's also zero. So, this is a zero divided by zero. So, at resonance, we need to deal with that. So, at resonance, we have x of t is going to be f times the limit that omega goes to omega naught of cosine omega t minus cosine omega naught t, divided by omega naught squared minus omega squared. The standard way of doing a limit like this when you have zero over zero is to use L'Hospital's rule. So, we have to differentiate the numerator and the denominator. We differentiate with respect to this variable which is omega, the limiting variable. We take the derivative of the numerator with respect to omega and divide it by the derivative of the denominator with respect to omega. This is equal to f times the limit that omega goes to omega naught. The derivative of the numerator with respect to omega. The derivative of cosine omega naught t will be zero. There is no omega there. The derivative of cosine omega t with respect to omega using the chain rule will pop out A t and then it will be a minus t sine omega t. All right. The derivative of cosine omega t with respect to omega. The derivative of the denominator with respect to omega, the derivative of omega naught squared is zero. So, it's the derivative of minus omega squared is minus two omega. At this point then, we can substitute in omega equal to omega naught. It's no longer zero over zero, and the minus signs cancel. So, we end up with a t times f from the front times sine omega naught t divided by 2 omega naught. That's the solution for our oscillator at resonance. What does it mean? The amplitude is t times f. So, it starts at zero but then t is growing in time. So, it's getting larger and larger. The vibration is getting larger and larger with each cycle and then a resonance, you get very large amplitudes which is why the wine glass breaks. Okay, let me review. I'm trying to model the physical problem where you have an oscillator being forced at the resonance frequency. The idea to think about is the wine glass has a natural frequency and we're singing at the same frequency at the wine glass as its natural frequency, trying to get the wine glass to vibrate violently and break. We model that in one dimension with this equation. It's a simple harmonic oscillator equation with a forcing term, a cosine forcing term. We know how to solve this equation, homogeneous solution plus particular solution. We can write down the general solution apply simple initial conditions where there is no motion and no displacement at t equals 0. Get the solution and then limit that solution at resonance, what happens when omega goes to omega naught, when the forcing frequency goes to the natural frequency. What we find is that the amplitude is increasing linearly with time getting larger and larger. I'm Jeff Chasnoff. Thanks for watching and I'll see you in the next video.