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Step 1: Understand the Initial Situation
• We have a pipe that is open at both ends, with an original fundamental frequency given as $f$.
• For an open pipe of length $L$, the fundamental frequency is given by
$f = \frac{v}{2L}$,
where $v$ is the speed of sound in air.
Step 2: New Configuration After Dipping in Water
• When the pipe is dipped in water so that half its length is submerged, the lower half is effectively sealed by water.
• Now, the air column has only the top half of the original length, which is $L/2$.
• This new arrangement acts like a pipe closed at the bottom (by water) and open at the top.
Step 3: Fundamental Frequency of a Closed Pipe
• The fundamental frequency of a pipe closed at one end (of length $L'$) is given by
$f' = \frac{v}{4L'}.$
• In our new situation, $L' = \frac{L}{2}$ (the length of the air column).
Step 4: Calculate the New Fundamental Frequency
• Substitute $L' = \frac{L}{2}$ into the closed-pipe formula:
$f' = \frac{v}{4 \times \left(\frac{L}{2}\right)} = \frac{v}{2L}.$
• Compare this with the original frequency:
$\frac{v}{2L} = f.$
Step 5: Conclusion
• The new fundamental frequency $f'$ matches the original frequency $f$.
• Therefore, even after dipping half the pipe in water, the fundamental frequency of the air column remains $f$.
