Question

A resonance tube of length 1m is resonated with a tuning fork of frequency 700 Hz. If the velocity of sound in air is 330 $m{{s}^{-1}}$ then the number of harmonics produced in the tube will be
A. 1
B. 2
C. 3
D. 4

Answer 1

Hint: First we will use the formula for the frequency of the n-th harmonic manipulate the expression to give us the number of harmonics when we have the frequency, length of the tube and the velocity of the wave in the given medium which in this case is the velocity of sound in air.

Formula used:
Frequency of nth harmonic
$\nu =\dfrac{(2n-1)v}{4L}$

Complete step by step answer:
We have the formula for the frequency of n-th harmonic and now we will manipulate it to give us the number of harmonics at the given frequency, length and speed of sound in air. We are taking the formula for pipe closed on one end because the water surface inside the pipe will act as a closed surface.
\[\nu =\dfrac{(2n-1)v}{4L}\Rightarrow n=\dfrac{\dfrac{4l\nu }{v}+1}{2}\]
We are given that the frequency of the sound produced by the tuning fork is 700 Hz. The length of the tube is given as 1 meter and the velocity of sound in air is 330 $m{{s}^{-1}}$. Putting all these values in the formula we get
\[n=\dfrac{\dfrac{4(1)(700)}{330}+1}{2}=4.74\]
The water level can be adjusted to produce different harmonics.

We will ignore the fraction value and take only the whole number part as only that many harmonics can be produced by varying the water level. So, we will have 4 harmonics in the given resonance tube. Hence, the correct option is D.

Note:
Take care that the system will be closed on one end and the corresponding formula must be used. The value after the decimal number should be ignored as the value of n can only be the whole number and the largest possible whole number value will be taken.