Frequency and String Tension (backwards)

[inflector]

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:

$$f \propto T^2$$

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?

 

[Studiot]

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?

I think I understand your question and the short answer is no.

If a string already has a tension imposed upon it by normal means (stretching) you don't significantly change that tension when you cause it to vibrate.

You noted a proportionality. Turn that into an equation with a constant call it A.

f = AT²

The constant, A, determines the amplitude and therefore the energy of the vibration.

Now each higher overtone requires more energy than the lower, with the fundamental requiring the least.

So the greater A is, the greater the energy input and the greater the overtone content.

Does this help?

 

[AlephZero]

When the string resonates at 2f, 3f, 4f, etc, the wavelength is 1/2, 1/3, 1/4, etc of the length of the whole string. The speed of the traveling wave is the same for all these frequencies and the tension is also the same.

If you excite a fairly large amplitude of vibration, you should be able to see the different vibrating shapes and the node positions where there is no motion of the string. At 2f there is a node at the mid point of the string. At 3f there are two nodes at 1/3 and 2/3 of the length, etc.

 

[timthereaper]

As far as I know, the string's vibration is somewhat quantized. You couldn't make the string vibrate (in standing wave fashion) with a sound generator at a frequency that isn't a harmonic of the natural frequency, which is defined by the tension.

 

[inflector]

Thanks for the replies so far.

What about my second case? What about a very long string before the wave is reflected and a standing wave arises? If you measure the tension at the point where the wave just reaches the accelerometer that should not be constrained by the harmonic frequency because there wouldn't be a standing wave absent reflection and interference.

Surely the tension must be higher in the case of a long string in space vibrating at a higher frequency under these circumstances? Mustn't it?

 

[Studiot]

Surely the tension must be higher in the case of a long string in space vibrating at a higher frequency under these circumstances? Mustn't it?

If a string already has a tension imposed upon it by normal means (stretching) you don't significantly change that tension when you cause it to vibrate.

I will try to rephrase this to make it more clear.

Unless the string has already been tensioned before vibration is attempted, waves of any character are impossible.

The tension in your and other formulae refer to this preset tension, not to any effect imposed by the vibration.

The derivation of the equation of wave motion on a string assumes that the string tension is not significantly altered in magnitude to a first order effect.

The action of a slack real string to vibrational input is governed by second order effects, since the first order ones are absent. You now have a forced oscillator with mechanical resistive losses as well as restoring forces. This is coupled to your mass. You have to solve the equations to predict the result.