Show on diagram,the fundamental mode of vibration,of the air column,in1.an open tube,2.Closed tube.If the frequencies of vibration are equal in the 2 tubes,find the ratio of their lenghts.

To visualize the fundamental mode of vibration in both an open tube and a closed tube, let's first understand how these tubes function as resonators. The fundamental mode, or the first harmonic, is the simplest form of vibration that occurs in these tubes. This can be illustrated with diagrams that represent the standing wave patterns formed within each type of tube.

Fundamental Mode of Vibration in an Open Tube

In an open tube, both ends are open to the air, allowing the air column to vibrate freely. The fundamental mode creates a standing wave with a node at each open end and an antinode in the center. The diagram below represents this concept:

• Node (N) at both ends
• Antinode (A) at the center

Fundamental Mode of Vibration in a Closed Tube

In a closed tube, one end is closed while the other end is open. This configuration results in a standing wave with a node at the closed end and an antinode at the open end. The diagram for this tube looks like this:

• Node (N) at the closed end
• Antinode (A) at the open end

Relationship Between Frequencies and Lengths

When the fundamental frequencies of vibration in both tubes are equal, we can derive a relationship between their lengths. The fundamental frequency of a tube is given by the formula:

• For an open tube: f = v / λ, where v is the speed of sound in air, and λ is the wavelength.
• For a closed tube: f = v / λ, but here, the wavelength is different due to the boundary conditions.

In an open tube, the wavelength (λ) for the fundamental mode is equal to twice the length of the tube (L): λ_open = 2L_open. For a closed tube, the wavelength is four times the length of the tube: λ_closed = 4L_closed.

Since the frequencies are equal, we can equate the two formulas:

v / (2L_open) = v / (4L_closed)

By simplifying this equation, we can cancel the speed of sound (v) from both sides, leading to:

4L_closed = 2L_open

From this, we can further simplify to find the ratio of the lengths:

L_open / L_closed = 2

Final Thoughts

This means that the length of the open tube is twice the length of the closed tube when both vibrate at the same fundamental frequency. This relationship highlights the differences in how standing waves are formed in tubes with different boundary conditions.