Phys 1240 Fa 05, SJP 2-1 Since I've started writing powerpoint lectures, it seems that writing detailed additional lecture notes (like I did for Chapter 1) may be superfluous. Here, just a brief summary and comments about the lecture on Chapter 2. (The powerpoints of individual lectures should serve as my lecture notes in the future!) But at least for this week, I'll continue with Key Chapter 2 points I want to emphasize: Chapter 2 zooms in on period and frequency. These refer to any periodic signal, something which repeats itself with a well defined period (time). The signal can be smooth and simple (a "perfect sine wave", like you see in Fig 2.12a) or a more complex but repetitive signal (like the ones in Fig 2.4 and 2.5 of the text) We already talked about these in the last chapter: Period is the time to get back to where you started, the time for a "cycle". Frequency is 1/Period, it's the inverse, the number of cycles in a second. (Frequency is measured in Hz, 1 Hz = 1 cycle/sec) Humans can hear periodic pressure waves from roughly 20 Hz to 20 kHz (kiloHz => 20,000 Hz) This depends, of course, on your age, the loudness, etc. The text goes on to write down one of the most important relationships in any study of waves, one which we will get to towards the *end* of covering Ch. 1 and 2, rather than the beginning. (Although I discussed wavelength in Ch. 1 notes, I've put it off in lecture to the end of the second week, that's why we won't see this formula until probably the end of week 2 ) It's the statement that wavelength * frequency = speed (for any wave) Or, in symbols, λ f = v It's important to understand WHY this formula is true. Try to picture a wave (a water wave might make the most direct visual image?)... You can think of it in various ways. E.g., you could zoom in on one point, where you will see the water bob up and down at the frequency f. Or, you can take a more holistic view, and watch the waves traveling along in front of you. (Focus on a "peak" or crest - it will move along in space, traveling outwards.) Or, you could take a snapshot - which does not pick one point in space, it picks one TIME. You will see a "wave" pattern, with peaks and troughs. The distance between one peak and the next is λ, the wavelengh. Now let the snapshot "run" like a movie - you will see the wave move, the crest will slide along... How long does it take for one crest to "slide over" one wavelength, λ? Think about it - it's exactly one period of the wave! (Watch the "destination spot" carefully, the place where the peak is heading to, the "next peak over". This spot starts high, but of course if you're focusing on ONE SPOT, it then goes low, and then gets high again... in exactly one period. That's how long it takes the wave to move a crest over to the next crest!) So the peak has moved a distance λ in one period. The speed it travels is Distance/time = λ/Period = λ * f. Again, v = λ f... This formula is generally true for any wave. They travel with a speed given by wavelength (distance from peak to peak) divided by period (TIME from peak to peak) For sound, v = 344 m/s at 20 C, and we discover that wavelength and frequency of sounds are related. If you wiggle something faster (higher frequency), the wave that you produce will have a smaller wavelength. Plug in numbers - for "typical, human" kind of frequencies (a few hundred Hz), the wavelength is on the order of meters, a "typical, human" kind of distance! (Coincidence? Maybe not, as we'll see when we think about how sounds are produced)Phys 1240 Fa 05, SJP 2-2 You can look at more complex waves, that aren't beautiful "sin" waves, and still they have a definite period, and they have a definite wavelength (the distance from peak to peak, or from any "equivalent point in the shape" to the next one...) And again, you'll find λ f = speed! Oscilloscopes are great tools for studying sound, because it's easy to build a microphone (a little flexible membrane that wiggles as the air pressure on the outside wiggles, and an electric sensor to measure that flex.) The pressure wave is thus converted to an electrical wiggle, and the scope shows the electric signal as a function of time. So you can "see" visually what the pressure is doing as time goes by. This image is called a "waveform". These are very characteristic - you can (with practice) recognize the waveform of a violin, an oboe, etc. We won't do that, although we will analyze waveforms more carefully to see what they tell us about the character of the sound. Generally, more complicated (but periodic) waveforms sound "richer" to our ears. (But not always, it's subtle! We'll come back to it) Vibrations generally have a period, they arise from a repetitive motion. Sometimes people call repetitive motion "harmonic motion" (a lovely name, which makes me think of music). The simplest kind of harmonic motion is the "sin wave" shown in Fig 2.12, this is called, naturally, "simple harmonic motion" (I'll abbreviate that SHM). The cool thing is that MANY OBJECTS which can vibrate tend to vibrate in this simple way! The smaller the vibration, the more closely most motion resembles SHM. It's a remarkable fact of nature, one which you can make sense of with a little more physics and calculus than I want to get into... but it's plausible. Look at vibrating objects: a whacked ruler, a swinging pendulum, a vibrating guitar string. They go back and forth smoothly and steadily, "position vs time" looks like Fig 2.12a! SHM arises if there is a force pulling the object "back to equilibrium" (back to the middle.) The whacked ruler "wants" to be straight. If it's bent up, the force on it pulls it down. If it's bent down, the force on it pulls it up. The farther it's bent, the MORE force there is pulling it back. If this "restoring force" (the force pulling it back to the middle) grows linearly with displacement (i.e, if by moving it twice as far away from equilibrium, it feels a force twice as big pulling it back), then the object will undergo SHM (Easily provable with a little more math than we need here, but if you like this kind of stuff, I'll point you to the chapter in any intro physics book that works out the details!) Here's an especially interesting, surprising, and important thing about SHM (simple harmonic motion). If you take such an object and whack it harder (so it starts off with MORE displacement away from the equilibrium, or "middle" point), what
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