Question 307: Waves

Question

A resonance tube closed at one end is of height 1.5 m . A tuning fork of frequency 340 Hz is vibrating above the tube. Water is poured in the tube gradually. The minimum height of water column for which resonance is obtained is (Neglect end correction, speed of sound in air $=340 \mathrm{~m} / \mathrm{s}$ )

Accepted Answer

When a resonance tube closed at one end is used, it vibrates in certain patterns or modes. For the tube to resonate in its first mode, the length of the air column above the water should be a quarter of the wavelength of the sound produced by the tuning fork.
Given:
Frequency of tuning fork, $ n = 340 \, 	ext{Hz} $
Speed of sound in air, $ V = 340 \, 	ext{m/s} $
Maximum height of the tube, $ 1.5 \, 	ext{m} $
In the first mode, the length of the air column, $ L_1 $, should be:
$  L_1 = \frac{V}{4n} = \frac{340}{4 	imes 340} = 0.25 \, 	ext{m} = 25 \, 	ext{cm}  $
This means the wavelength $ \lambda $ for the first mode is:
$  \lambda = 4 	imes L_1 = 4 	imes 25 \, 	ext{cm} = 100 \, 	ext{cm}  $
Resonance occurs not only at $ \frac{\lambda}{4} $ but also at higher harmonics where the length is $ \frac{3\lambda}{4}, \frac{5\lambda}{4}, \frac{7\lambda}{4} $, etc.
Calculating for $ \frac{5\lambda}{4} $:
$  L = \frac{5 	imes 100}{4} = 125 \, 	ext{cm}  $
Since the height of the tube is 150 cm (1.5 m), resonance is achieved when:
The air column length is 125 cm.
Thus, the height of the water column needed for this setup is:
$  	ext{Height of water} = 	ext{Total tube height} - 	ext{Air column length} = 150 \, 	ext{cm} - 125 \, 	ext{cm} = 25 \, 	ext{cm}  $
Therefore, the smallest height of water column for resonance is 25 cm.

Answer

75 cm

Answer

50 cm

Answer

30 cm

Answer

25 cm

Waves

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