Question

A tuning fork of frequency 480 Hz resonates with a tube closed at one end of length, 16 cm and diameter 5 cm in fundamental mode. Calculate velocity of sound in air.

Answer 1

Hint: The velocity of a sound wave is equal to the product of the frequency of the sound and the wavelength of the sound. We have the value of frequency and the wavelength can be calculated from the fact that the resonance is taking place in the fundamental mode of vibration for which the wavelength is related to the dimensions of the tube.
Formula used:
The relation between the velocity, the frequency and the wavelength of sound waves is given as
\[v = \nu \lambda \]
In the fundamental mode of vibration, the wavelength of the sound is related to the length and the diameter of the tube by the following relation:
$\dfrac{\lambda }{4} = l + 0.3d$

Complete answer:
We are given a tuning fork which is vibrating at the frequency given as
$\nu = 480Hz$
It is given that this fork resonates with a tube closed at one end. The length of this tube is given as
$l = 16cm = 0.16m$
The diameter of the tube is given as
$d = 5cm = 0.05m$
The resonance is occurring in the fundamental mode of vibration and the relation for wavelength in terms of the dimensions of the tube in fundamental mode is given as
$
  \dfrac{\lambda }{4} = l + 0.3d \\
  \lambda = 4\left( {0.16 + 0.3 \times 0.05} \right)m \\
  \lambda = 0.7m \\
 $
Now we have the values of the wavelength and the frequency of the sound waves. We can calculate the velocity of light in the following way.
\[v = \nu \lambda = 480 \times 0.7 = 336m/s\]
This is the required velocity of sound.

Note:
1. The standing waves are produced in the given tube and the sound waves are vibrating in one loop which is known as the fundamental mode of vibration and is the simple possible mode.
2. The relation for velocity holds for electromagnetic waves as well. In that case, we talk about the velocity of light which has the fixed value of $3 \times {10^8}m/s$.