# Can someone explain how the frequency of a string relates to the tension?

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

• en.wikipedia.org/wiki/String_vibration – G. Smith Jan 22 '20 at 3:37
• dimensional analysis – JEB Jan 22 '20 at 4:46
• thanks for no help guys – Milan Pin Jan 22 '20 at 5:37
• @MilanPin Using standing waves. – aditya_stack Jan 22 '20 at 6:25

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