One dimensional sound waves explained
Spinola puts the record straight again
My colleagues' somewhatropey grasp of physics
resulted in a deluge of emails pointing out the error of our ways. But when we apologised for the error, a second torrent of mails arrived, this time from readers as bemused by all this stuff as we are, asking us to publish an explanation for one-dimensional sound waves.
Eager as ever to please, here's just two of the many dissertations we received.
David Robertson offers: "One dimension doesn't mean you have to have zero magnitude. Imagine having a Slinky spring laid out on the ground, with one end fixed. Now stretch it out to maybe six feet long.
"If you quickly push the end you're holding towards the fixed end, and then pull it back again, you'll see a compression wave scoot its way along the Slinky to the fixed end.
"Note that you now have a 'wave' travelling in a single dimension (more or less) along the Slinky's length.
Any more exciting and you'd fall outta your chair, eh? :)"
And if you want more detail, Walton Comer's your person: "As any physicist would tell you, it is usually best to question everything you hear or read until you have convinced yourself of the truth in it. So I will try to help explain how a sound wave can have an amplitude and carry energy while remaining confined to motion in a single dimension. It is almost identical to have binary data move through a single wire on a data bus.
"The simplest example is of a plucked guitar string. In this case the vibrating medium (the string) is one-dimensional. No doubt, someone might point out that in fact a guitar string moves in at least a two-dimensional space, which is true. However, mathematically, the string remains a one-dimensional creature, as it's topology never changes.
"Perhaps a more compelling example would be that of a one-dimensional compression wave. A compression wave is one in which the motion of energy is parallel to the wave-like motion. In the example of the guitar string, the energy travels in the direction of the unplucked string, while the wave-like motion lies perpendicular.
"The sound waves travelling through air are examples of compression waves. Compression waves exist as a series of high and low pressure regions. Air from the high pressure regions moves quickly into the low pressure regions, so quickly that the previously high pressure regions, become low pressure. Analagous to a stream of ones and zeroes moving through a data bus.
"The wavelength is the separation between two high pressure regions (two 1's). The amplitude is the difference between the high and the low pressures (the difference between 0 and 1, could be 1.5 volt, 5 volt, maybe 12 volts.)
"In open air, these waves normally travel in all three dimensions, but, if confined to the interior space of a hollow tube, they would effectively travel in only a single dimension (the smaller the tube, the more true this becomes.)
"As a side note, if the sound was confined to moving between two very large flate sheets, lying side by side, then it would be travelling effectively in only two dimensions. In the case of the carbon nano-tubes, the interior space is to small for air to effectively carry a sound wave; instead they must travel through the walls of the tube.
"You might imagine the air confined between two concentric tubes (one lying within another.) In this case, the compression wave would be able to travel in two separate dimensions, as long as the radial dimension is large with respect to the wavelength.
"If we take a step back from the tubes, it should be clear that the energy is only being directed in the direction that the tubes lies, as their are barriers to its travelling outside the tubes. So already it is clear that the energy is confined to travelling in only on direction.
"Furthermore, if the radial dimension is smaller than the wavelength then, the motion of the compression is effectively confined to the direction in which the tubes lie, as well. (Back of the envelope - the wavelength is around one decimeter, give or take an order of magnitude, while the radial dimension is at least 5 orders of magnitude less than this.)
"For the carbon nano-tubes these sound waves are travelling through the carbon itself, not air, but the principle is the same."
So now you know. ®