Question

The vibration from an 800Hz tuning fork setup standing waves in a string clamped at the both ends the wave speed in the string is known to be 400m/s for the tension used. The standing wave is observed to have four antinodes. How long is the string?

Answer 1

Firstly calculate the length of string by using the formula of frequency of oscillation of the string in the ${n^{th}}$ harmonic there are four antinodes in the string put the value of n as 4 because it is ${4^{th}}$ harmonic.

Complete answer:
We know it is given that the frequency of oscillation of the string is 800Hz and speed of the wave is known to be 400m/s. using the formula, frequency of oscillation of string in ${n^{th}}$ harmonic is $\dfrac{{nc}}{{2l}}$
Where ‘c’ is the speed of the wave in string, ‘l’ is the length of the string and ‘n’ is the number of harmonics. Thus, $f = \dfrac{{nc}}{{2l}}$
Put the values of f, c in this formula we get,
$800 = \dfrac{{n \times 400}}{{2l}}$
Since, there are four antinodes in the string n = 4
Put n = 4 in the above expression
\eqalign{ & 800 = \dfrac{{4 \times 400}}{{2l}} \cr & \Rightarrow 2l = \dfrac{{4 \times 400}}{{800}} \cr & \Rightarrow l = \dfrac{{4 \times 400}}{{2 \times 800}} \cr}
After solving this we get as:
$\therefore l = 1m$
Therefore the length of string is 1m.

Additional information:
A tuning fork serves as a useful illustration of how a vibrating object can produce sound. The fork consists of a handle and two times when the tuning fork is hit with a rubber hammer the times begin to vibrate. The back and forth vibration of tines produce disturbances of the surrounding air molecules. Standing wave is also known as a stationary wave, in which the peak amplitude will remain constant with time but the wave is oscillating in time.

Note:
To calculate the length of the string which is an oscillating formula, the frequency of oscillation of string should be known to us as there are 4 antinodes in this question. So it becomes ${4^{th}}$ harmonic whereas antinodes are positions on standing waves where the wave vibrates with the maximum amplitude.