Lectures on Physics has been derived from Benjamin Crowell's Light and Matter series of free introductory textbooks on physics. See the editorial for more information.... 
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Standing Waves
The photos below show sinusoidal wave patterns made by shaking a rope. I used to enjoy doing this at the bank with the pens on chains, back in the days when people actually went to the bank. You might think that I and the person in the photos had to practice for a long time in order to get such nice sine waves. In fact, a sine wave is the only shape that can create this kind of wave pattern, called a standing wave, which simply vibrates back and forth in one place without moving. The sine wave just creates itself automatically when you find the right frequency, because no other shape is possible.
If you think about it, it's not even obvious that sine waves should be able to do this trick. After all, waves are supposed to travel at a set speed, aren't they? The speed isn't supposed to be zero! Well, we can actually think of a standing wave as a superposition of a moving sine wave with its own reflection, which is moving the opposite way. Sine waves have the unique mathematical property that the sum of sine waves of equal wavelength is simply a new sine wave with the same wavelength. As the two sine waves go back and forth, they always cancel perfectly at the ends, and their sum appears to stand still. Standing wave patterns are rather important, since atoms are really standingwave patterns of electron waves. You are a standing wave!
Standingwave patterns of air columnsThe air column inside a wind instrument or the human vocal tract behaves very much like the waveonastring example we've been concentrating on so far, the main difference being that we may have either inverting or noninverting reflections at the ends.
Inverting reflection at one end and uninverting at the other
If you blow over the mouth of a beer bottle to produce a tone, the bottle outlines an air column that is closed at the bottom and open at the top. Sound waves will be reflected at the bottom because of the difference in the speed of sound in air and glass. The speed of sound is greater in glass (because its stiffness more than compensates for its higher density compared to air). Using the type of reasoning outlined in optional section 4.2, we find that this reflection will be densityuninverting: a compression comes back as a compression, and a rarefaction as a rarefaction. There will be strong reflection and very weak transmission, since the difference in speeds is so great. But why do we get a reflection at the mouth of the bottle? There is no change in medium there, and the air inside the bottle is connected to the air in the room. This is a type of reflection that has to do with the threedimensional shape of the sound waves, and cannot be treated the same way as the other types of reflection we have encountered. Since this chapter is supposed to be confined mainly to wave motion in one dimension, and it would take us too far afield here to explain it in detail, but a general justification is given in the caption of the figure. The important point is that whereas the reflection at the bottom was densityuninverting, the one at the top is densityinverting. This means that at the top of the bottle, a compression superimposes with its own reflection, which is a rarefaction. The two nearly cancel, and so the wave has almost zero amplitude at the mouth of the bottle. The opposite is true at the bottom  here we have a peak in the standingwave pattern, not a stationary point. The standing wave with the lowest frequency, i.e. the longest wave length, is therefore one in which 1/4 of a wavelength fits along the length of the tube.
Both ends the sameIf both ends are open (as in the flute) or both ends closed (as in some organ pipes), then the standing wave pattern must be symmetric. The lowestfrequency wave fits half a wavelength in the tube.


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