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 take a sine wave and make a copy of it shifted over, their sum is still a sine wave. The same is not true for a square wave.(PSSC Physics)

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 standing-wave patterns of electron waves. You are a standing wave!

(PSSC Physics)

Standing-wave patterns of air columns

The air column inside a wind instrument or the human vocal tract behaves very much like the wave-on-a-string 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

Surprisingly, sound waves undergo partial reflection at the open ends of tubes as well as closed ones. The reason has to do with the readjustment of the wave pattern from a plane wave to a spherical wave. If the readjustment was as sudden as that shown in the figure, then there would be kinks in the wave. Waves don't like to develop kinks. In section 4.2 we deduced the strength of the reflected wave at a change in medium from the requirement that the wave would not have discontinuities or kinks. Here there is no change in medium, but a reflected wave is still required in order to avoid kinks.

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 density-uninverting: 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 density-uninverting, the one at the top is density-inverting. 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 standing-wave 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 same

If 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 lowest-frequency wave fits half a wavelength in the tube.

Self-Check

Draw a graph of pressure versus position for the first overtone of the air column in a tube open at one end and closed at the other. This will be the nextto- longest possible wavelength that allows for a point of maximum vibration at one end and a point of no vibration at the other. How many times shorter will its wavelength be compared to the frequency of the lowest-frequency standing wave, shown in the figure? Based on this, how many times greater will its frequency be? Graphs of excess density versus position for the lowest-frequency standing waves of three types of air columns. Points on the axis have normal air density.

Answer

The wave pattern will look like this: Three quarters of a wavelength fit in the tube, so the wavelength is three times shorter than that of the lowest-frequency mode, in which one quarter of a wave fits. Since the wavelength is smaller by a factor of three, the frequency is three times higher. Instead of fo , 2 fo , 3 fo , 4 fo , ..., the pattern of wave frequencies of this air column goes fo , 3 fo , 5 fo , 7 fo , ...