Phys 1240 Fa 05, SJP 8-1 DRAFT Chapter 8: Sound Spectra We just learned that if you wiggle a string, only certain special frequencies, the "harmonics", will resonate. All other frequencies tend to damp away almost instantly, so if you pluck a string, you will hear the fundamental frequency f1, AND the higher harmonics (2f1, 3f1, 4f1, 5f1, etc) all at the same time, all superposed. If you look back to Chapter 2 (section 2.2, on waveforms), we showed pictures of complex waves that had some basic (fundamental) frequency, but didn't look like a simple sin wave. Here's one of the most remarkable things about waves (discovered by a mathematician named Fourier): If you superpose a bunch of waves (ALL harmonics of a given fundamental frequency f1) you can generate ANY POSSIBLE waveform that repeats itself at frequency f1. In other words, all those complex waveforms shown back in Chapter 2.2. of the text can be thought of as arising from some particular simple sum of harmonics of the basic frequency f1. A given string will produce a particular characteristic sum of harmonics (e.g., the higher harmonics might die away quickly in amplitude), and that will yield a "sound", or "timbre" associated with the string. You perceive the fundamental frequency f1 as the "pitch", and all the higher harmonics add in just to change the shape of that tone from the whiny, annoying pure sine wave to a richer, fuller tone. The point is, if a string plays A440, you hear a PITCH of 440 Hz, but the character of the sound depends on how many (and which, and how strong) harmonics you add in. If you don't add in any, you hear a very "electronic" sound. If you add in higher harmonics in just the right way, you can produce an "A" that sounds like a violin, or a saxophone, or... just about anything! The difference is not in the frequency, but in the mix of higher harmonics! By the way, if you add in "random" frequencies, that is, frequencies which are not an exact multiple of the fundamental (something between f1 and 2f1, for example) then you get "anharmonic" sound. It CAN be interesting, some drums and bells have this kind of sound, it generally doesn't have a very clear pitch associated with it. It can also sound pretty bad, like "noise". If you play the piano and play two notes which are right NEXT to each other, they "clash", they are dissonant. In part, that's because the frequencies of one are not related in the simple, nice way that harmonics are. One other cool thing about our hearing. If I play 300 Hz and 600 Hz together, they "fit". The 600 Hz is a higher harmonic (n=2) of the first. It's an octave. If I play, say, 300 Hz and 900 Hz together, it sounds nice too. 900 Hz is NOT the same note (it's not an octave, or two octaves, but someplace in between), but because it's a higher harmonic (n=3), they still sound very nice together. If I play, say, 300 Hz and 450 Hz together, we again get something nice. In this case, 450 is NOT a harmonic at all. But BOTH of these tones are harmonics of a common, lower, fundamental (150 in this case. 300 is twice, and 450 is three times, this other fundamental). Now things are getting really interesting! You play 300 and 450, and your brain sort of vaguely senses 150, which isn't there but you "feel" its presence. Part of the reason is that 150 Hz signals repeat every 1/150 of a second. So do 300 Hz signals! They repeat every 1/300 of a second, which means you're ALSO back to where you started after 1/150 sec. Similarly, 450 Hz signals repeat every 1/450 sec, which means your back to where you started (for the third time) after 1/150 sec. For some reason, we seem to like hearing sounds that repeat themselves! If you play 300 and 320 Hz together, it doesn't sound so nice. It's fairly dissonant. (This is about like playing two notes right next to each other on a piano. Kind of jarring). You could argue that these are still both harmonics of a common frequency, but that common frequency is going to be 20 Hz, VERY far away from either one.Phys 1240 Fa 05, SJP 8-2 DRAFT Section 8.1 of the text starts off by showing you various wave forms in Figure 8.1, all of which (except the last) have the exact same frequency. (They all repeat themselves a little more than three times in the time shown in the graph) The details are totally different, one is a "square", the other a "pulse", the other a "triangle" (the first is a pure sine) but they will all SOUND like the exact same frequency. They just have a different timbre. They are each BUILT UP by superposing a bunch of different harmonics of that same (common) fundamental frequency. The mathematics here is a little fierce, but also doesn't matter so much. We will play with this in the Phet simulation (go to http://www.colorado.edu/physics/phet/web-pages/simulations-base.html and click on "Fourier: making waves) where you can see how to build up complicated waves by adding in different strengths of harmonics. I strongly suggest you go play with this yourself! None of the waves in Figure 8.1 of the text sound especially nice. They all sound "artificial", computer-like. A little harsh to my ears, anyway. Waves with "sharp edges" (like Fig 8.1c) have lots of higher frequencies, lots of "high harmonics" (People also call these "overtones"). It's a funny thing - you are adding up smooth waves, and yet you can build up these sharp edges.( The sim will help you see how that works!) When you add a bunch of harmonics, you need to be careful to note two things about every frequency that you add in. First, the AMPLITUDE. That's just the strength. So e.g. if you start with a pure sin wave of frequency f1, and amplitude 1 (in some units, maybe you're talking about pressure waves so this is 1 N/m^2 overpressure). Then you could add in a sin wave of frequency 2f1, and you can choose any amplitude you want. If you have an amplitude much BIGGER than 1, it will tend to dominate. If you add in this higher harmonic with amplitude much SMALLER than 1, it will subtly alter the timbre of the fundamental. In most natural instruments, this is how it goes automatically - higher harmonics tend to have smaller and smaller amplitudes. There is another thing you can alter, the "phase" of the wave. Remember that just means shifting the sin wave left or right. So if two waves are "in synch", they both start at zero at a common time. But if you shift one of them, if you change its "phase", then one wave might e.g. be zero when the other is not...
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