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