A stationary wave is set up in the tube with the same frequency as the tuning fork. The lower end of the tube is sealed.

Question 11

The diagram shows a tuning fork above a tube of air of length 25 cm.

A stationary wave is set up in the tube with the same frequency as the tuning fork. The lower end of the tube is sealed. This is the minimum length of tube with the lower end sealed that creates a stationary wave.

Which other lengths of tubes, sealed at their lower end, will also create a stationary wave?

A 37.5 cm and 50 cm

B 50 cm and 75 cm

C 75 cm and 100 cm

D 75 cm and 125 cm

Reference: Past Exam Paper – November 2015 Paper 12 Q28

Solution:

Answer: D. 

When a stationary wave is formed, a node will always be at the closed end and an antinode will be at the open end.

So, the simplest stationary wave formed will have a node at the closed end and an antinode at the open end [this is known as the fundamental mode]. This corresponds to a quarter of a wavelength.

λ / 4 = 25 cm

Wavelength λ = 25 × 4 = 100 cm

The next stationary wave is formed as shown [this is the 1^st overtone].

This is ¾ of a wave corresponding to the next length L of the tube.

3λ / 4 = L

Since λ = 100 cm,

L = 3 × 100 / 4 = 75 cm

The second overtone consist of one and a quarter of a wave (= 5λ / 4).

5λ / 4 = L

L = 5 × 100 / 4 = 125 cm

As seen, the longer tubes supporting a stationary wave must have nodes and antinodes in the patterns NANA (75 cm), NANANA (125 cm), etc.