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I am investigating Mersenne's law with a guitar by varying tension (hanging weights) and string length. Will temperature change (room temperature to ~4°C) effect the frequency noticeably? If so, is the string oscillating differently or is the change due to a variation of the speed of sound? The strings have a free end so the contraction of the string will not increase tension. Any help would be appreciated.

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The resonance length will change because of the temperature change as well as the speed of sound change. –  Mr.ØØ7 May 30 '13 at 8:08
    
Know any useful related formulas? –  Nicolas May 30 '13 at 8:53
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2 Answers

The speed of sound according to this chart would change as much as 330.4/358.0 = 8% . The air behaves closely to an ideal gas, therefore this kind of change would hardly change the pitch, let alone be noticeable to a human ear.

However the change in temperature would effect the stiffness and length of the string dramatically, this depends on what material you are using. Look at the difference of length vs temperature.

Since you are using weights, your stress is not going to change, and the tension would remain the same.

Try using this equation:

Equation of frequency of a string

But this is not the full picture, since you are actually using an open string with dampening. Though with those speeds it should be approximated to a closed string.

You can try using: $$f_open = \frac{\sqrt{\frac{T}{m/2}}}{2L} \cdot constant$$

And tune the constant to fit your current parameter space.

Source with calculator (The calculator though would not work in your case because one of your ends is open).

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The frequency does not depend on the medium of propagation. The wavelength does. That is why I think that the temperature influences on the frequency of oscillations only due to changes in elasticity or length of the string.

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