Cup and string theory

Sep 13, 2006 4:45 am
Since two cups and a string make up one of the most primitive transmission systems, folks at free103point9 have long wondered about the most efficient cup and string for communication. Years ago, our co-conspirator Jason Wilson submitted this question to The Mad Scientist Network and got this answer from Arnd Pralle Post-doc/Fellow, Mol. Cell Biol., UC Berkeley:

What are the best (most resonant) materials for a string and cup phone?

I am no expert in acoustics or material science, however since it a nice question showing the difficulties to get a practical estimate from basic physical principles, I have tried to find an answer based on physics which can be found in any general textbook.

Let's look at the two parts of the string and cup phone:
The string transmits the vibrations by oscillating laterally. To do this best, it has to fulfill the following:

The elasticy along the string (E) should be minimal while the lateral flexibility (G) maximal. The properties of solids, including metal wires do not show a very large G/E ratio (steel or iridium wires would be best).

However polymers would be much better suited for the task. They should be thin to avoid damping by the surrounding air and it should be lightweight, to minimize the tension necessary to hold it up.

So the best bet seems to be a polymer, e.g. nylon string. I would recommend using a kite-string, the thinnest one available.

The cup of a string and cup phone collects the acoustic wave which causes the bottom of the cup to vibrate. Therefore it is a good idea to have a truncated cone-shaped cup with rigid (thicker) sides that reflect the sound wave onto the bottom. The bottom has to be resonant. The top (opening) of the cup should fit the size of the users head for easiest use. The bottom size should relate to the frequency of human language and the most sensitive range of our ears which peaks around 500-1500 Hz. Depending on the bottom's design, I would estimate a diameter of 8-15cm. Just think about your medium speakers. When one looks at speakers or drums, it is clear that a thin membrane would be the best resonantor. However, the resonance frequency of a membrane depends largely on the lateral tension. Because in the phone the membrane has to support the wire, this tension could not be optimized for the frequency but must support the string. Therefore a thin plate is a much more suitable bottom resonator. Again, similar considerations as for the string apply: either a thin steel foil/sheet would be good, or again a polymer bottom plate seems optimal. Actually, the use of yogurt cups does seem to be a rather smart choice.

Special considerations for outside installation: weather resistance (thermal expansion is so small in normal temperature range that it can be ignored).
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