Both of the above strings have their ends locked in place. The first string, $S_1$, is twice as long as the second string, $S_2$. If sound waves are going to be sent through both, what is the correct ratio of the fundamental frequency of $S_1$ to the fundamental frequency of $S_2$?
A) $2:1$
B) $\sqrt{2}:1$
C) $1:\sqrt{2}$
D) $1:2$
A) $2:1$
B) $\sqrt{2}:1$
C) $1:\sqrt{2}$
D) $1:2$
A wave passing along a string with both ends held in place will have a fundamental frequency of
$$f=\displaystyle \frac{v}{(2l)}$$
where $v$ is the speed of the wave and $l$ is the length of the string. Thus, fundamental frequency is inversely proportional to the length of the string, so the first string, which is twice as long as the second, will have a fundamental frequency that is half of the second string’s. Expressed as a ratio, that is $1:2$.