This question deals with the fundamental frequency of an air column in a cylindrical tube when its boundary conditions change. Let's break it down step by step.

Step 1: Understand the Initial Condition (Open-Open Tube)

A tube open at both ends is called an open-open tube. For such a tube, the simplest standing wave pattern (fundamental mode) has a node at the center and antinodes at both open ends. The length of the tube, L, is equal to half the wavelength ($\lambda$).

Therefore, the relationship is: $L=\frac{\lambda }{2}$

We can find the wavelength: $\lambda =2L$

The fundamental frequency (f) is given by the wave equation: $f=\frac{v}{\lambda }=\frac{v}{2L}$, where $v$ is the speed of sound in air.

Step 2: Analyze the New Condition (Half-Dipped in Water)

When the tube is dipped vertically in water so that half of it is submerged, the lower end becomes closed (by the water surface), while the upper end remains open to the air. The tube now behaves like a closed-open tube. The length of the air column is now half of the original length, i.e., $L\text{'}=\frac{L}{2}$.

Step 3: Find the New Fundamental Frequency

For a closed-open tube, the fundamental mode has a node at the closed end and an antinode at the open end. The length of the tube is equal to one-quarter of the wavelength ($\lambda \text{'}$).

Therefore, the relationship is: $L\text{'}=\frac{\lambda \text{'}}{4}$

We can find the new wavelength: $\lambda \text{'}=4L\text{'}=4\times \frac{L}{2}=2L$

The new fundamental frequency (f') is given by: $f\text{'}=\frac{v}{\lambda \text{'}}=\frac{v}{2L}$

Step 4: Compare the Two Frequencies

Notice that the expression for the new frequency is identical to the original frequency: $f\text{'}=\frac{v}{2L}=f$

Final Answer: The fundamental frequency remains the same, f.

Related Topics & Formulae

Standing Waves in Pipes:

• Open-Open Tube: Fundamental wavelength, $\lambda =2L$. Fundamental frequency, $f=\frac{v}{2L}$.
• Closed-Open Tube: Fundamental wavelength, $\lambda =4L$. Fundamental frequency, $f=\frac{v}{4L}$.

The key is to correctly identify the boundary conditions (open or closed) to determine the possible standing wave patterns.