|MadSci Network: Physics|
First: I doubt you can induce actual sub harmonic vibrations in your ring. You might be able to get it to vibrate in two or more different frequency modes at once and produce an audible beat frequency (the difference between the two interfering frequencies) that is lower than the fundamental. This is one of the things that can happen when you tap the ring because it often excites several frequencies at once. Moreover since the ring is not actually vibrating at this beat frequency you can’t use it as a sub harmonic drive frequency to put the ring into resonance. These beat frequencies will also not show up in a frequency spectrum even though your ear perceives them as such. Second: Because your steel ring is almost certainly a very high quality resonator (it has very little damping and so rings for quite a while) you can probably get it to resonate by using a sub harmonic drive frequency. This is like pushing a kid on a swing every other time or every third time and so on. The swing still resonates at the same frequency as always, you just drive it at less than its natural frequency. Third: You should be able to get a pretty good frequency spectrum experimentally using a microphone and a freeware program like “spectrogram 5.0” or an earlier release, by Richard Horne. (I think he has a souped up shareware version out now too.) You will most likely find a fundamental at about 11 or 12 KHz. A really high pitched ping but certainly audible. This is because you are exciting transverse waves when you tap the ring and the proper speed of sound to use for them is about 3200m/s. (See www.efunda.com/materials/ ). The 6000m/s speed of sound is for longitudinal (compression) waves, the kind of wave we usually associate with sound in air or water. Most metals though are about half as stiff in shear as they are in tension so you get two different speeds depending on how you stress them. You may in fact notice slightly different frequencies for bending of the ring in plane (when you tap its rim inward toward the center) and out of plane (when you tap it perpendicular to the plane in which it lies) because its stiffness is different in these two directions due to its shape. Lastly: The ring may not have “ends” like a pipe or string but since you can’t send longitudinal waves around it in only one direction you’ll still get constructive interference every 180degrees and a fundamental frequency just as you calculated. It would probably be difficult to hear a pure longitudinal vibration because the ring wouldn’t change overall shape very much and thus not displace very much air to produce a sound. In fact it may be difficult to excite just longitudinal waves because they involve tension and compression of the metal tangent to the ring. The stick slip from rubbing tangentially, like a wet finger on the rim of a glass, might seem like it would work but in fact it ends up exciting transverse waves too. As you have noted supporting the ring is a bit tricky and can affect which frequencies you get, but suspending it from strings horizontally or vertically should do the trick. You could also rest it horizontally on a few narrow supports. Thanks for an intriguing question. I hope this answers some of it.
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