Question

An open pipe is suddenly closed at one end with the result that the frequency of third harmonic of the closed pipe is found to be higher by \(100 \mathrm{~Hz}\) than the fundamental frequency of the open pipe. What is the fundamental frequency of the open pipe? (A) 240 (B) 200 (C) 300 (D) 480

Short answer

The fundamental frequency of the open pipe is 200 Hz. The correct choice is (B) 200.

Step-by-step solution

Recall the formulas for the fundamental frequency of open and closed pipes.

The formula for the fundamental frequency (1st harmonic) of an open pipe is: \(f_o = \dfrac{v}{2L}\) Where \(f_o\) is the fundamental frequency of the open pipe, \(v\) is the speed of sound, and \(L\) is the length of the pipe. The formula for the fundamental frequency (1st harmonic) of a closed pipe is: \(f_c = \dfrac{v}{4L}\) Notice that the frequency of the 3rd harmonic of a closed pipe is three times the fundamental frequency: \(3f_c = \dfrac{3v}{4L}\) Now, we are given that the frequency of the 3rd harmonic of the closed pipe is 100 Hz higher than the fundamental frequency of the open pipe: \(3f_c = f_o + 100\)

Express the fundamental frequency of the open pipe in terms of the closed pipe frequencies.

We can replace \(3f_c\) and \(f_o\) in the frequency relationship equation using the formulas for the fundamental frequency of open and closed pipes: \(\dfrac{3v}{4L} = \dfrac{v}{2L} + 100\)

Solve the equation for the fundamental frequency of the open pipe.

To solve for \(f_o\), divide both sides of the equation by \(v\): \(\dfrac{3}{4L} = \dfrac{1}{2L} + \dfrac{100}{v}\) Next, subtract \(\dfrac{1}{2L}\) from both sides of the equation: \(\dfrac{1}{4L} = \dfrac{100}{v}\) Now, multiply both sides by \(4L\) : \(1 = \dfrac{400L}{v}\) Divide both sides by 400: \(\dfrac{1}{400} = \dfrac{L}{v}\) Finally, substitute the formula for the fundamental frequency of the open pipe: \(f_o = \dfrac{v}{2L}\) Since \(\dfrac{1}{400} = \dfrac{L}{v}\), the equation becomes: \(f_o = \dfrac{1}{\dfrac{1}{200}}\) Which simplifies to: \(f_o = 200\) So, the fundamental frequency of the open pipe is 200 Hz. The correct choice is (B) 200.

Key concepts

Open Pipe Harmonics

Open pipe harmonics are a fascinating concept that involves sound waves resonating inside an open pipe. In an open pipe, sound waves travel back and forth, creating standing waves.
The fundamental frequency, or first harmonic, occurs when the pipe supports a single full wavelength. For open pipes, the formula to find this fundamental frequency is given by:

• \( f_o = \dfrac{v}{2L} \)

Where:

• \( f_o \) is the fundamental frequency
• \( v \) is the speed of sound in the medium
• \( L \) is the length of the pipe

For open pipes that have both ends open, all harmonics are present because standing waves can form full wavelengths perfectly. Other harmonics, like the second harmonic (first overtone), third harmonic (second overtone), etc., are perfectly integer multiples of the fundamental frequency.
Each harmonic results in a distinct frequency that contributes to the musical tone produced by the pipe.

Closed Pipe Harmonics

Closed pipe harmonics have their unique characteristics due to having one end closed and the other open. This alteration changes the vibration patterns compared to open pipes. Instead of forming full wavelengths, closed pipes form nodes at the closed end and antinodes at the open end.
The fundamental frequency or first harmonic for a closed pipe is given by:

• \( f_c = \dfrac{v}{4L} \)

Here:

• \( f_c \) is the fundamental frequency
• \( v \) is the speed of sound
• \( L \) is the length of the pipe

Importantly, only odd harmonics (1st, 3rd, 5th, etc.) are present in closed pipes. Full even harmonics cannot be formed in closed pipes because they require an additional node at the closed end, which is not possible.
This characteristic results in a different set of tonal qualities compared to open pipes, heavily influencing its application in musical instruments and acoustics.

Frequency Relationships

Understanding frequency relationships is fundamental in analyzing sound waves in pipes. The key is the relationship between different harmonics and how they interact.
In the exercise, we explored how the third harmonic of a closed pipe relates to the fundamental frequency of an open pipe. Specifically: * The third harmonic of a closed pipe is three times its fundamental frequency: \( 3f_c = \dfrac{3v}{4L} \). * The fundamental frequency of an open pipe: \( f_o = \dfrac{v}{2L} \). In this case study, the third harmonic of the closed pipe was 100 Hz higher than the fundamental frequency of the open pipe:

• \( 3f_c = f_o + 100 \)

Through solving these relationships, we deduced the open pipe's fundamental frequency. Such frequency relationships are critical in tuning instruments and understanding their harmonic properties. They ensure the desired sound qualities by coordinating the fundamental and overtone frequencies.

Acoustics

Acoustics is the branch of physics that explores the study of sound. In the context of pipes, acoustics helps in understanding how sound waves behave as they resonate inside tubes, whether they are open or closed.
Sound waves in pipes can create harmonics due to resonance, and understanding these acoustic principles is crucial for designing musical instruments, architectural acoustics, and noise control systems.
Acoustics relies on the principles of:

• Wave motion: Understanding how sound waves travel and interact with the pipe's physical structure.
• Resonance: How the natural frequencies of the pipe amplify sound waves, creating clear tones.
• Damping: Consideration of energy loss within the medium or pipe structure.

By studying acoustics, one can understand not only how harmonics function in pipes but also how these principles apply to broader sound-related phenomena. From orchestra halls to pipe organs, acoustics plays a pivotal role in shaping how we experience sound.