Question

The fundamental frequency of a guitar string is 367 hz . part a what is the fundamental frequency if the tension in the string is halved?

Answer 1

The fundamental frequency of a string is given by:
$f_1 = \frac{1}{2L} \sqrt{ \frac{T}{\mu} }$
where L is the string's length, T the tension and $\mu$ the linear density of the string.

We can see that f1 is proportional to the square root of T: $\sqrt{T}$.
This means that if the new tension is half the initial value, the new fundamental frequency will be proportional to $\sqrt{ \frac{T}{2} }= \frac{ \sqrt{T} }{ \sqrt{2} }= \frac{f_1}{ \sqrt{2} }$

So, the new fundamental frequency will be
$f_1 ' = \frac{367 Hz}{ \sqrt{2} }=259.5 Hz$