Question

When an air column of length $l$ closed at one end produces its fundamental note, the wavelength is given by:
A) $\lambda = l$
B) $\lambda = 2l$
C) $\lambda = 4l$
D) $\lambda = 3l$

Answer 1

Hint: In order to solve this question, proper knowledge of what is a wavelength and how it is calculated is required, also the conditions of the open and closed end of the air column is necessary because when the air column is open an antinode is formed but when the column is closed a node is formed. Keep in mind whether an antinode is forming or node and solve the question accordingly.

Complete step by step answer:
It is given that the air column is closed at one end, so it can be simply assumed that the air column is open at one end.
So, we have the condition in which one end is closed and one end is open.
At the closed end of the air column a node is formed and at the open end of the air column an antinode is formed.
So, the distance between node and the successive antinode is given by, $\dfrac{\lambda }{4}$.
This is equal to the length of the air column,
$\dfrac{\lambda }{4} = l$
So, the expression for wavelength would become,
$\lambda = 4l$

Hence, option C is the correct answer.

Note: In this question the concept of standing waves is used. Standing Waves are generally associated with resonance in air columns and it is described mainly in terms of displacement of air in columns. These waves can also be visualized in terms of the pressure variations in the air column.
A node for a displacement is always an antinode for the pressure and a node for the pressure is always an antinode for a displacement. When air is constrained to a node, the air motion will be alternately squeezing towards that point and expanding away from it, causing the pressure variation to be at the maximum.