For the lesson this week I'd like to spend a little bit of time talking about resonance. we're going to talk about the resonance of strings. we're going to talk about the resonance of cavities and we're going to lead into a discussion that talks about reverberation in a room. we'll go from a discussion of standing waves. to a discussion of the the natural decay of rumacoustic's, that's really based on statistical energy. the interesting thought is that you play an instrument within an instrument. So a guitar has Is a stringed instrument that basically can be tuned and create musical notes. And actually, that instrument interacts with the room that you're in. And the room itself becomes a part of the signature of the sound, and so at the end of this lesson you should understand a little bit about the, the, the pieces that correspond to that part of the music production. So standing wave and room acoustics are going to be the focus here. I'm going to start again with the vibrating string. And you know, I'd like to consider a string that's fixed at both ends meaning that the displacement, sorry about that, the displacement is zero here. And it's also zero here at x equal l. So this is the dimension x. And of course we have a displacement of the string. It can vibrate in the y direction here as a function of the spatial position x and time. If you study partial differential equations, you would know how to derive the solution to the response. it's done through a process known as separation of variables. we're not going to cover that here I'm going to jump straight to the solution. for the string, the response of the string, which is represented here. And you can see that, that it is sinusoidal in the x dimension and that we also have a harmonic response represented here with the complex exponential. And Professor Bako has a a discussion on the complex exponential but it's and I'll let him continue with that but the point is this is a harmonic response, actually is a complex exponential is represented by sine and cosine functions. So we basically have a wave number K which you see here and K is defined in terms of the overall length of the string here and it takes on various integer values. So the solution that you see here is for N equal 1, 2, 3. And on and on. There are an infinite number of solutions or wave forms that fit between the boundaries of the string as it's plucked or driven in vibration. In the time domain, there is a natural frequency this is the, what is known as the circular natural frequency. It's 2 pi times the natural frequency, which is represented by Fs of N here. And it is related to the length of the string and the speed of sound, and the string with following relationship. Where N is the integer pi. C divided by the length. And you can see here, that C is defined in terms of the tension that's in the string. And this is the mass per unit length as of N is a corresponding amplitude, and then I mentioned T is the tension in the string. So c, our speed of sound, is represented by the square root of the ratio, the tension in the string to the mass per unit length. So, as I mentioned, there are many modes that define the response. In fact, there's an infinite number and we can represent the response of this, of the vibrating string by the expression here for n equal 1, 2, 3. Again, you, you see the the harmonic behavior the relationship here in the time domain expressed in terms of amplitudes. Sine and cosine, all multiplied by the spatial response, that's also harmonic in nature, as well. At any given instant in time, the response of the string is defined as the sum of the response of all of the modes. Meaning for all possible wave forms. So at n equal 1 this is the wave form that corresponds to n equal 1. for n equal 2 this is the standing wave or wave form that corresponds to n equal 2. And similarly, for n equal 3 this is the way form corresponds. And this goes on and on. And you can see that for n equal 3, there are two modes, meaning two points where the response is zero. For n equal 2, there's one. for any equal 1 there's 0 between the boundaries where the response is 0. and these are the wave numbers, k equal 1, k equal 2 and k equal 3, corresponding to the indices. And that defines the wave numbers are defined as. Pi on l, two pi on l for n equal two and three pi on l for n equal three.