- #1
inflector
- 344
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I'm trying to figure out how much an increase in the frequency of vibration of a string increases the tension in the string. NOTE: This is not homework, I'm not in school.
I know that the fundamental resonant frequency and harmonics are a function of string tension because string tension changes the rate of propagation of waves in the string and therefore the wavelength and frequency. What I'm trying to figure out is a different question and perhaps the reverse.
How much does the tension in a string change as the frequency of a wave goes up. For example, if you create a resonance of the same amplitude at f, 2f or 3f where f is the fundamental frequency, how much does the tension on the string change?
I note here:
http://hyperphysics.phy-astr.gsu.edu/Hbase/waves/string.html
under the section: Vibrating String Frequencies, what is sometimes called the Second Law of Vibrating Strings, that the doubling the frequency requires quadrupling the tension since:
[tex]f \propto T^2[/tex]
But is this valid in the other direction. If one drives a string using acoustic vibrations, for example, so that it vibrates at 2f where f is its fundamental, does that increase the tension on the string to 4T?
A related question is what would happen with a very long vibrating metal string in an inertial frame in space where you tied one end to an accelerometer equipped mass, and hooked the other up to an electromagnetic vibrator that oscillated transverse to the string. If you started two experiments, A with the tension at zero and a frequency of 1Hz, and B with the tension at zero and a frequency of 2Hz. Assume that the mechanical vibrator puts enough energy into each string so that the string itself vibrates at the desired frequency and a constant amplitude between the A and B experiments. What would be the difference in measured acceleration just after the tension increase between the A and B experiments?
I know that the fundamental resonant frequency and harmonics are a function of string tension because string tension changes the rate of propagation of waves in the string and therefore the wavelength and frequency. What I'm trying to figure out is a different question and perhaps the reverse.
How much does the tension in a string change as the frequency of a wave goes up. For example, if you create a resonance of the same amplitude at f, 2f or 3f where f is the fundamental frequency, how much does the tension on the string change?
I note here:
http://hyperphysics.phy-astr.gsu.edu/Hbase/waves/string.html
under the section: Vibrating String Frequencies, what is sometimes called the Second Law of Vibrating Strings, that the doubling the frequency requires quadrupling the tension since:
[tex]f \propto T^2[/tex]
But is this valid in the other direction. If one drives a string using acoustic vibrations, for example, so that it vibrates at 2f where f is its fundamental, does that increase the tension on the string to 4T?
A related question is what would happen with a very long vibrating metal string in an inertial frame in space where you tied one end to an accelerometer equipped mass, and hooked the other up to an electromagnetic vibrator that oscillated transverse to the string. If you started two experiments, A with the tension at zero and a frequency of 1Hz, and B with the tension at zero and a frequency of 2Hz. Assume that the mechanical vibrator puts enough energy into each string so that the string itself vibrates at the desired frequency and a constant amplitude between the A and B experiments. What would be the difference in measured acceleration just after the tension increase between the A and B experiments?