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chapter10-1

# chapter10-1 - Physics 1240 Hall Chapter 10.1 Notes 1 Focus...

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Physics 1240 Hall Chapter 10.1 Notes 1 Focus questions and learning goals Chapter 10.1 focus questions: 1. How does a stringed instrument work? 2. How is the sound affected by the physical characteristics of the string and the way it is played? Chapter 10.1 learning goals. After studying this chapter, you should be able to: 1. Predict what frequencies a vibrating string can produce, and how they are related to one another. 2. Describe what properties of the string can be used to change the pitch (and which don’t matter). 3. Describe and quantify the impact of various changes you might make like increasing tension, swapping the string for a more massive one, or a longer one. 4. Look at a snapshot of a vibrating string and describe the characteristics, including wavelength and pitch. 5. Explain how the end conditions determine the allowed frequencies of a string. 6. Predict what happens to the music if you put your finger lightly (or heavily) at various points on a string. We skipped ahead to this section because it contains an essential piece of physics for under- standing harmonics and timbre . We’ve already talked about it some, but not in a lot of detail. It’s the idea of a standing wave on a string, like we discussed back in Chapter 4. 2 Speed of a wave on a string If you wiggle a string, of course a traveling (transverse) wave will propagate. It will have a certain speed which is given by the formula v = radicalBigg tension mass per unit length . We haven’t derived this formula (although it looks a little like the formula for the frequency of oscillation of a mass on a spring!) Let’s see if we can make sense of it. It says that if the tension gets bigger, then waves travel faster. Seems reasonable—we’ll do this in class with rubber tubing, and with a slinky. More tension allows the wiggling material to “send the message” to the next piece of material that much easier and faster. The signal travels faster! The fact that mass is in the downstairs also seems reasonable—heavy materials will be sluggish, waves should travel more 1

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Physics 1240 Chapter 10.1 notes slowly. The square root is not very obvious, but if you think carefully about the units, you’ll see that you need it. (I leave that as a puzzle for the ambitious—tension has units of force (which is mass times acceleration). Can you see that the units really do work out, after taking the square root, to be meters/second? 3 Standing waves If the string is clamped at both ends (like a string on any musical instrument), then when you send waves down, they will bounce off the end, come back, and superpose with the waves going the other way. We saw (with animations) that when you superpose two traveling waves going in opposite directions, you get a standing wave. Why is it called “standing”? Well, it’s not obviously moving left or right any more. If you stare at it, you just see string bobbing up and down, there is no longer any sense of left or right motion. You can understand it as the superposition of a left-going and a right-going wave: those back and forth waves add up to build a standing wave. In
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