Question

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 the nth harmonic. Choose the correct option. (A) \(n=3, f_{3}=(3 / 4) f\) (B) \(n=3, f_{2}=(5 / 4) f_{1}\) (C) \(\mathrm{n}=5, \mathrm{f}_{2}=(5 / 4) \mathrm{f}\) (D) \(\mathrm{n}=5, \mathrm{f}_{2}=(3 / 4) \mathrm{f}_{1}\)

Short answer

(B) \(n=3, f_{2}=\frac{5}{4}f_{1}\)

Step-by-step solution

Calculate second harmonic frequency of the open pipe

The formula for the frequency of any harmonics in an open pipe is: \(f_n = \frac{nV}{2L}\) Where \(f_n\) is the frequency of the n-th harmonic, V is the speed of sound in air, and L is the length of the pipe. Given that the open pipe is in resonance with its second harmonic, \(f_1 = \frac{2V}{2L}\) and from this, we have \(f_1 = \frac{V}{L}\)

Calculate the harmonic frequency of the closed pipe

The formula for the frequency of any odd harmonics in a closed pipe is: \(f'_n = \frac{(2n-1)V}{4L}\) The closed end pipe resonates at its nth harmonic, so \(f_2 = \frac{(2n-1)V}{4L}\)

Solve for the relation between f1 and f2

Divide the equation of the closed pipe by the equation of the open pipe: \(\frac{f_2}{f_{1}} = \frac{(2n-1)V}{4L} \times \frac{L}{V}\) Simplifying the equation, we get \(\frac{f_2}{f_{1}} = \frac{2n-1}{4}\) We need to find the value of n for which the above equation holds true. Solution (contd.):

Check given options

Let's test the given options: (A) \(n=3\) and \(f_{3}=(3 / 4) f\) We are given that n should be equal to 3. However, the frequency is incorrect due to f3 being written instead of \(f_2\). Let's calculate the frequency if n = 3, \(\frac{f_2}{f_{1}} = \frac{2(3)-1}{4} = \frac{5}{4}\) Option (B) fits this condition, and thus the correct choice is: (B) \(n=3, f_{2}=(5 / 4) f_{1}\)

Key concepts

Open Pipe Harmonics

Open pipes produce harmonics that include a series of sound frequencies that occur at integer multiples of the fundamental frequency. The fundamental frequency is known as the first harmonic. In physics, harmonics refer to specific patterns of standing waves that develop inside a pipe due to resonance. Open pipe harmonics occur because sound waves reflect back and forth within the pipe, creating nodes and antinodes at successive intervals along the pipe's length.

For an open pipe, both ends support an antinode due to the boundary conditions, which results in an open pipe supporting all integer harmonics: 1st, 2nd, 3rd, and so forth. The frequency of the first harmonic for an open pipe is given by the equation:

\[ f_n = \frac{nV}{2L} \]

Here, \( f_n \) is the frequency of the \( n \)-th harmonic, \( V \) is the speed of sound, and \( L \) is the pipe's length. The open pipe's ability to produce all harmonics is crucial in musical instruments like flutes and organ pipes.

Closed Pipe Harmonics

In contrast to open pipes, closed pipes have one end closed and exhibit a different pattern of resonance and harmonics. Closed pipes only support odd harmonics, which leads to a distinctive sound character. The presence of a closed end creates a node (point of zero amplitude) at that end, while there is an antinode (point of maximum amplitude) at the open end.

As a result, the resonance pattern means that a closed pipe will produce harmonics at frequencies described by odd integer multiples of the fundamental frequency. The mathematical relationship for the harmonic frequencies of a closed pipe is:

\[ f'_n = \frac{(2n-1)V}{4L} \]

Here, \( f'_n \) stands for the frequency of the \( n \)-th harmonic (where \( n \) represents the sequence of odd numbers 1, 3, 5, etc.), demonstrating why a closed pipe doesn't produce even harmonics. This feature is utilized in some wind instruments, such as clarinets, contributing to their unique sound.

Frequency in Resonance

Resonance is a phenomenon that occurs when the frequency of an external force matches the natural frequency of a system, leading to maximum energy transfer and amplified oscillation. In the context of pipe resonance, it refers to the specific frequencies at which a pipe will naturally vibrate and produce sound.

Both open and closed pipes have specific resonant frequencies dictated by their harmonic structures, which affect how they sound when air is blown through them. Understanding the concept of resonance in pipes helps in accurately tuning musical instruments to achieve desired tones.

When dealing with changing scenarios such as transitioning an open pipe to a closed configuration, the frequency relationships change. For example, as explained in the original problem, changing from an open pipe to a closed pipe involves recalculating the harmonics and corresponding frequencies. The relationship between the original frequency and the new adjustment is governed by:

\[ \frac{f_2}{f_{1}} = \frac{2n-1}{4} \]

This ratio helps determine which harmonic a closed pipe will reach, given a certain frequency, making it a powerful tool for sound analysis and instrument design.