Question

A pipe of $30\,cm$ long and open at both the ends produces harmonics. Which harmonic mode of pipe resonates at $1.1\,kHz$ source?(Given speed of sound in air $ = 330m{s^{ - 1}}$)
A) Fifth harmonic
B) Fourth harmonic
C) Third harmonic
D) Second harmonic

Answer 1

Hint:Here we have to apply the concept of standing sound waves.
A standing sound wave, also known as a stationary wave, is a wave that oscillates in time but whose peak amplitude profile does not shift in space. On average, there is no net distribution of energy from waves of similar magnitude moving in opposite directions.

Complete step by step solution:
When two similar waves pass in opposite directions down a line, they form a stationary wave-that is, a wave form that does not travel through space or along a line even though it is made up of two waves that migrate in the same direction.
We may create a standing wave in a tube that is open on both sides, and in a tube whose one end is open and one end is closed. Open and closed ends reflect waves in a different way. The closed end of the tube is the antinode in the pressure of the tube. The open end of the tube is around a node in the pressure.
The longest standing wave in the L-long tube with two open ends has a displacement of antinodes at both ends. It is called the fundamental or the first harmonic. The next longest standing wave in an L-long tube with two open ends is the second harmonic. It also has antinodes of displacement at each end. The whole number of half wavelengths must fit into the L-long tube.
Given,
$v = 330\,m{s^{ - 1}}$
$L = 30\,cm = 0.3\,m$
$\nu = 1.1\,kHz = 1100Hz$
The frequency of ${n^{th}}$ harmonic in an open pipe
$
{\upsilon _n} = \dfrac{{nv}}{{2L}} \\
\Rightarrow 1100 = \dfrac{{n \times 330}}{{2 \times 0.3}} \\
\Rightarrow n = 2 \\
$
Which is the second harmonic.

Note:Here we have to see the unit of the length while calculating the frequency. The unit should be standard like the speed of sound.