Waves

To determine the length of the closed pipe, we need to compare the frequencies of the harmonics of both the closed and open pipes. Given:
- Frequency of the fifth harmonic of a closed pipe equals the frequency of the third harmonic of an open pipe.
- Length of the open pipe, . Step 1: Determine the Frequencies For a closed pipe, the frequency of the
-th harmonic is: where is an odd integer (1, 3, 5, ...). For an open pipe, the frequency of the
-th harmonic is: where is any positive integer (1, 2, 3, ...). Step 2: Set the Frequencies Equal Given that the fifth harmonic of the closed pipe equals the third harmonic of the open pipe: Substitute the expressions: Step 3: Solve for Cancel from both sides: Rearrange to solve for : Substitute : Final Answer: This corresponds to option (1).

Step 1: Understanding the Fundamental Frequency of an Open Pipe
The fundamental frequency of an open pipe is given by: where is the speed of sound and is the length of the pipe.
Step 2: Effect of Closing One End and Partially Filling with Water
- When one end is closed, the pipe behaves as a closed pipe, where the fundamental frequency is: which is half of the open pipe's frequency.
- If the bottom of the pipe is filled with water, the effective length of the vibrating column becomes .
Step 3: New Fundamental Frequency Calculation
Since the effective length is reduced, the new fundamental frequency becomes: Given that Hz, we substitute: Step 4: Conclusion
Thus, the new fundamental frequency of the pipe is Hz.

Beats are given by: Since , we have: So, could be 476 Hz or 484 Hz. When the tension in A is increased, the frequency of A increases. This means: But since f was initially either 476 or 484, the correct answer is 484 Hz.

The speed of a wave on a string is given by: where is the tension and is the mass per unit length of the string. The wave speed increases by 20\%, so the new speed is: The ratio of the new speed to the original speed is: The speed is proportional to the square root of the tension, so we have: Squaring both sides: Thus, the new tension is: The percentage increase in tension is: Thus, the percentage increase in tension is 44\%.