I've already spoken about, standing waves. but, for each mode it's, it's, it's referred to frequently as a standing wave. And there's a relationship, between the, wave number, the frequency and the speed of sound, and that's expressed here. And remember, the frequency which we defined earlier, the natural frequency this is the expression for the natural frequency. So all I've done is substituted for f's of n here in the expression. And if we do that and solve for the wave length, we get an expression for the wave length that's defined as 2L divided by N. And so notice if you remember on the diagram in the previous slide, the, the, the fundamental mode of the, of the string, looked basically like this. And so, the wave, the full wavelength actually will continue on outside the boundary of the string. And you can see that here in the expression. So when n equals l the wavelength is twice the length of the string which is defined here. Okay? And so we get the, the, the appropriate wavelength for n equal 1, n equal 2, n equal 3 through the expression here. so that's, that's the wavelength we've previously sketched. it's also worth noting that, that, in the case of the string here, that f2. The, second natural frequency, is equal to 2 times the first. And the third natural frequency is equal to 3 times the first, etcetera. and these are called overtones because they're integral multiples, of the fundamental. And this is [INAUDIBLE] useful for stringed instruments, like guitars. where the notes can be changed. I'm going to talk a little bit about the guitar strings in particular. I've, I've sketched out the the frets for a typical neck of a guitar here. And, you know, what we've shown here is basically where one might place your fingers on the strings. To to create a chord. And, of course, these are the frets associated with the guitar. And we can change the length of the string, the tension stays fixed, the mass-per-unit length stays fixed. But we can change the tension, I'm sorry, the length of the string. And when we do that it's going to change the frequency. And I can demonstrate that as well. >> So, given that this is a course in audio music engineering, it seems important to bring the the discussion on the engineering side back to the musical side. And we've been talking about strings under a fixed level of tension. And if you remember the speed of sound was related to the tension in the string and the mass per unit length. And of course, you know, on an instrument like the guitar or other instruments the diameters of the strings are typically different. And that obviously changes the mass per unit length. turns out, you know, for the guitar I've got this in a standard tuning, so the first and sixth string are both tuned to E. but if you remember, if we change the length of the string, then the frequency changes in inverse proportion. So if we were to decrease the length of the string, we're effectively going to increase the the frequency associated with the resonance of the string itself. So, for example, here's the first string open [SOUND] and then I can apply my finger on the fret board [SOUND]. And you immediately hear the change in the pitch, and of course that's the key [SOUND] to a tuned instrument is that [SOUND] we actually have the ability to change the frequency [SOUND] fret board. And you know in music we call that our guitar chords, and so we, that's what turns the theory of strings into music. [MUSIC] Just one last comment on the changing the length of the strings. We obviously have bar chords but, you know, there is a, a device that we use called the capo, that we can apply to the The neck of the guitar. And basically, change the, length of all the strings at the same time. So that's a quick way for us to, change the overall frequency of all the strings uniformly. >> There's a really nice, website, that's constructed by, Professor Dan Russell. at Penn State and he teaches in the Graduate Acoustics program. But if you follow the link to the web site on your own there are many different animations that are displayed there. I'm going to focus a few because he covers the natural modes of a fixed-fixed string as you can see here on the screen. And if you you know all of the equations that he represents here are consistent with what we talked about already. But I thought you might like to see an animation of the modes. So this is N equal 1, N equal 2, N equal 3, N equal 4. And you can see the standing waves that this is the motion of those each of those modes as they vibrate. Now the key point I wanted to make earlier was is that the total response of the string at any given time is the summation of the response of all of these individual modes together. So you can see here vi, the vibration of a fixed string that is plucked at a distance 1 third of its length. And so the black line here is the total response of the string frozen in, in one instant of time. what you see here, is the first mode, the second mode, the third and the fourth modes. And basically if you were to sum the response of all of these modes together. You begin to see this general motion or behavior. So one forumlates the sum of the response of the individual modes to actually construct the actual picture of the reponse of the string itself. So this is a really good webpage I would encourage you to to look at it through there are a number of different areas of to to study the response characterisitcs.