An open pipe is in resonance in \(2^{\text {nd }}\) harmonic with frequency \(\mathrm{f}_{1}\). Now one end of the tube is closed and frequency is increased to \(\mathrm{f}_{2}\) such that the resonance again occurs in nth harmonic. Choose the correct option. (A) \(\mathrm{n}=5, \mathrm{f}_{2}=(5 / 4) \mathrm{f}_{1}\) (B) \(\mathrm{n}=3, \mathrm{f}_{2}=(3 / 4) \mathrm{f}_{1}\) (C) \(\mathrm{n}=5, \mathrm{f}_{2}=(3 / 4) \mathrm{f}_{1}\) (D) \(\mathrm{n}=3, \mathrm{f}_{2}=(5 / 4) \mathrm{f}_{1}\)

For an open pipe, the resonance formula says that when a pipe is in resonance, the frequency is given by \[f = \frac{2n}{L} \times v\] where n is the harmonic number, L is the length of the pipe, and v is the speed of sound.

We are given that the open pipe is in resonance in the 2nd harmonic with frequency f1. Using the formula from Step 1, we can write the frequency equation for the open pipe: \[f_1 = \frac{4}{L} \times v\]

For a closed pipe, the resonance formula says that when a closed pipe is in resonance, the frequency is given by \[f = \frac{2n - 1}{4L} \times v\] where n is the harmonic number, L is the length of the pipe, and v is the speed of sound.

We are given that the closed pipe is in resonance in the nth harmonic with frequency f2. Using the formula from Step 3, we can write the frequency equation for the closed pipe: \[f_2 = \frac{2n - 1}{4L} \times v\]

Now we need to find a relationship between f1 and f2 using the equations from Steps 2 and 4. We can do this by dividing the equation of f2 by the equation of f1: \[\frac{f_2}{f_1} = \frac{(2n - 1)/4L}{4/L}\]

After simplifying the relationship from Step 5, we have: \[\frac{f_2}{f_1} = \frac{2n -1}{16}\]

Now that we have a relationship between f1 and f2, we must check the provided options to find the correct one. For this, we need to plug in the given values of n and f2/f1 into the relationship from Step 6. Let's first plug the values from Option (A): \(n = 5, f_2 = (5/4)f_1\) \[\frac{5}{4} = \frac{2(5) - 1}{16} \implies \frac{5}{4} = \frac{9}{16}\] This is not correct. Now plug the values from Option (B): \(n = 3, f_2 = (3/4)f_1\) \[\frac{3}{4} = \frac{2(3) - 1}{16} \implies \frac{3}{4} = \frac{5}{16}\] This is correct. There is no need to check more options as we have already found the correct answer. The correct option is (B). So, n = 3, and the relationship between f1 and f2 is \[f_2 = \frac{3}{4}f_1\]