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and come to the conclusion that the The fundamental frequency of a vibrating string is directly proportional to the square root of the tension using equations please. This is for a lab i'm doing on guitar strings

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    $\begingroup$ en.wikipedia.org/wiki/String_vibration $\endgroup$ – G. Smith Jan 22 '20 at 3:37
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    $\begingroup$ dimensional analysis $\endgroup$ – JEB Jan 22 '20 at 4:46
  • $\begingroup$ thanks for no help guys $\endgroup$ – Milan Pin Jan 22 '20 at 5:37
  • $\begingroup$ @MilanPin Using standing waves. $\endgroup$ – aditya_stack Jan 22 '20 at 6:25
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The wave speed can be related to the tension and the mass per unit length of the string by the following equation:

$$ v = \sqrt{\frac{T}{\mu}}$$

Here, $T$ is the tension, $\mu$ the mass per unit length and $v$ the speed of waves on the string. For a derivation of this equation refer to this or to any first year Physics textbook (e.g. Halliday & Resnick).

In the fundamental vibrational mode of the string, the two the ends are half a wavelength apart. In the $n$th harmonic, the length $l$ of the string is $\frac{n \lambda}{2}$. Thus, the wavelength $\lambda$ of the string is $\frac{2l}{n}$. Apply the wave equation:

$$ v = nf_{1} \frac{2l}{n} \implies \sqrt\frac{T}{\mu} = 2lf_{1} \implies f_{1} = \frac{1}{2l}\sqrt\frac{T}{\mu}$$

Here's an image to explain what's going on

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