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2 April 2004
Sound Velocity from Standing Wave Resonance
John W. Dooley, Physics Department
Millersville University
This month's experiment requires a small localized sound source. It is best to use tiny earphones and caution to avoid subjecting your ears to excessively loud sounds.As we saw in the previous installment ( http://www.sas.org/E-Bulletin/2004-02-20/ ), identical tones from two separate sources can be summed to produce a louder or a quieter net sound, depending on how the two sound waves arrive at the ear. For a louder sound, the crests from both waves should arrive at the same time, a condition known as constructive interference. This happens when the separation between the two sources is a whole number of wavelengths.
In this month's experiment we create constructive interference using only a single source and a reflector. The reflected sound acts as though it came from a second source, much as the reflected light in a mirror acts as though it came from an object behind the mirror. A schematic diagram of the idea is shown below.
In the sketch, the reflected wave travels an extra 6 wavelengths before reaching the ear. Because the extra travel distance is a whole number of wavelengths, the reflected wave crests arrive at the ear at the same time as the direct wave crests. The result is constructive interference and loud sound, just as if we had set up two speakers in the previous exercises, 6 wavelengths apart.
The schematic diagram does not quite represent this month's experiment. One difference is that for us the source and the reflectors will all lie along the same line. In the diagram, they are separated vertically to make the idea clearer.
One version of the actual experiment is shown below. A soda can is emptied and the tab is bent inside and out of sight. A small earphone speaker is placed facing down into the hole in the top of the can. (The other earphone of the set in the photograph is not used.) Sound generated by the speaker bounces back and forth between the top and the bottom of the can.
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For most sound wavelengths, the reflected waves do not interfere constructively with the original wave. Only for wavelengths which "fit" inside the can will the reflected sound exhibit constructive interference with the speaker sound. When this happens, the sound wave in the can will grow to a large amplitude, and we will hear loud sound for that wavelength. When this happens, we say that the sound is resonant within the can.
To achieve resonance, we adjust the sound wavelength to find a "fit" by adjusting the frequency of the signal that excites the loudspeaker. (Remember that the speed of sound = frequency * wavelength .) The figure below shows the GoldWave setup for this experiment. Only one channel has sound. The other channel is undriven, so only one earphone is excited (the one in the can).
The green graph represents a 200 Hz sine wave, playing in a loop. A large sound file may be downloaded for this experiment, here .
The black and white graph represents the distribution of frequencies for that signal, which is called its spectrum. The spectrum peaks at about 670 Hz instead of 200Hz. 670 is the frequency of the signal actually being sent to the speaker.
The difference between 670Hz actual and 200Hz original is created by playing the digital instructions faster than normal. The play-back speed is controlled by the bottom (of 3) slide bar, shown between the green graph and the black graph. The frequency of the signal from the speaker is raised by "grabbing" the slide indicator with the mouse and dragging it to the right. You can also adjust the playback rate in steps by clicking on the - or + to the left or right of the bar.
The soda can is about 0.1 meter high, so that sound with a wavelength 0.2 meter should ring strongly. The reflected waves travel an extra wavelength each time and add constructively with the speaker wave.
Earlier articles ( http://www.sas.org/E-Bulletin/2003-11-14/features/index.html and http://www.sas.org/E-Bulletin/2004-02-20/features/index.html ) determined a sound speed of about 350 m/sec, so a 350 Hz tone should have a wavelength of 1 meter. Because the speed of sound = frequency * wavelength, we expect the 670 Hz signal to have a wavelength of about 0.5 meter.
A more careful measurement of the can height still leaves us with a problem to understand. The can is about half as high as it "should be" to make one round trip equal to a wavelength. It is as though two round trips are required to make the reflected wave add constructively with the speaker wave. The wave model for sound has worked so well up to now that we seek a story which has this two-round-trip feature.
It turns out that the first figure above, representing two reflections, is correct for sound pressure, provided that each reflection is from a hard wall. Air piles up against the wall. The pressure at the wall then rises, and a high pressure region is reflected away. For the soda can, at the hole where the speaker is mounted, the story is different.
Instead of piling up at the end, air rushes out through the hole. An arriving high pressure region rushes out, overshoots, and leaves behind a low pressure region at the opening. This low pressure region creates the reflected wave. At an open hole, the reflected wave is inverted . The wave reflecting from the bottom of the can is not inverted--the low pressure region reflects as a low pressure region.
Because of this single inversion, a wave that travels an extra wavelength as it travels to the top of the can combines destructively with the speaker wave. Making one round trip equal to a wavelength is exactly wrong if constructive interference is what we want.
We can get constructive interference after one round trip if the speaker is producing low pressure when the reflected wave arrives. This happens when the speaker is half-way through its cycle when the reflected wave arrives. But that means that the round-trip time is only half a period, and the round trip distance is only half a wavelength.
The upshot is this: When one end is open and one end closed, the can is resonant when its length is 1/4 of a wavelength. According to this model, we should also find resonance when the can is 3/4, 5/4/, ... etc. ... of a wavelength. We predicted no resonance when the can length is equal to 1/2 or 1 wavelength.
Because the top of the can includes some "hard wall," it might be possible to get resonance when the length is a wavelength. To test the model, we can use a tube with a more clearly open end, as in the photograph below. That tube is 2.54 cm (1-inch) diameter acrylic about 1 meter long. It is part of a standard resonance experiment in which the length of the resonant air tube is varied by pouring water into the tube. (The water surface acts as the closed end of the tube.)
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Click image to enlarge |
If we fill the tube until the air column is 0.12 meter long (from water to rim of tube), we expect a resonance at about 700 Hz, when the wavelength is about 0.5 meters and1/4 wavelength is 0.123 meters. We adjust the frequency until we hear the resonance. Now we push the model and the experiment:
We drain the tube slowly and listen for the next resonance (when the air column length is 3/4 of a wavelength). When we do, we expose a small surprise: The length of the second air column (from water to top of tube) "should" be 3 times the length of the original resonant length, but it is not.
The reason is an ambiguity in the length of the tube. When the high pressure air rushes out, it initially moves straight up, dispersing a little above the top of the tube. The effective length of the tube is longer than the measured length. Dr. Conrad R. Miziumski of our department at Millersville University pointed out to me that, even though the effective length is hard to determine, the change in air column length between the two resonances is exactly 1/2 of a wavelength.
The most accurate measurement of the sound wavelength (at a given frequency) is twice the distance between water levels for two adjacent resonances. Knowing that wavelength and the frequency gives us an accurate determination of the speed of sound inside the tube.
As a final test, consider this
question: What will be the resonant wavelengths if both
ends of the tube are open? 