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Chapter 4

Resonance of a string

Three pure tones of frequencies 12 Hz, 4 Hz, and 3 Hz are played one at a time near a taut string of length 1 m that is tied at both ends. It is observed that only the 12 Hz tone makes the string vibrate in one of its allowed modes. From this information, can you deduce
the LNRF of the string? The wave velocity for the string? All the allowed frequencies?

 

Answer:

It is clear that 4 Hz and 3 Hz Hz are not resonant frequencies of the string since those tones do not excite the string. However, this does not necessarily imply that 12 Hz is the LNRF (lowest natural resonant frequency). For example, it is possible that 6 Hz is the LNRF of the string. That would fit the data that the 12 Hz tone does cause the string to vibrate, but the 4 Hz and 3 Hz tones do not. However, there are other possibilities as well: Suppose that the lowest mode is 12/5 Hz. (The lowest mode does not have to be an integer number of Hertz!) Then 12 Hz = 5(12/5) Hz would still be a resonant frequency, but 4 Hz and 3 Hz would not (since there is no integer multiplying 12/5 that gives 4 or 3). We are unable to tell from the information given what the LNRF is. We have no information on the wave velocity either. We know that 12 Hz, 24 Hz, etc, are allowed frequencies, but we do not know what others are possible.

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