Solved Examples of Harmonics in Open and Closed Pipes

📚 Quick Study Guide

• 📏 Open Pipes: Both ends are open, allowing maximum displacement.
• 🎵 Harmonics in Open Pipes: The frequencies are integer multiples of the fundamental frequency ($f_n = n \frac{v}{2L}$), where $n = 1, 2, 3, ...$
• 💨 Closed Pipes: One end is closed (node) and one end is open (antinode).
• 🎼 Harmonics in Closed Pipes: Only odd harmonics are present ($f_n = n \frac{v}{4L}$), where $n = 1, 3, 5, ...$
• 📢 $v$: represents the speed of sound in the medium (usually air).
• 📝 $L$: represents the length of the pipe.

Practice Quiz

1. A pipe is open at both ends and has a length of 1.5m. If the speed of sound in air is 343 m/s, what is the frequency of the fundamental mode?
   1. 114.3 Hz
   2. 228.7 Hz
   3. 343.0 Hz
   4. 514.5 Hz
2. A pipe closed at one end has a length of 0.75m. Using a speed of sound of 343 m/s, calculate the frequency of the fundamental mode.
   1. 57.2 Hz
   2. 114.3 Hz
   3. 228.7 Hz
   4. 457.3 Hz
3. An open pipe has a fundamental frequency of 440 Hz. What is the frequency of its second harmonic?
   1. 220 Hz
   2. 660 Hz
   3. 880 Hz
   4. 1320 Hz
4. A closed pipe has a fundamental frequency of 500 Hz. What is the frequency of its third harmonic?
   1. 1000 Hz
   2. 1500 Hz
   3. 2000 Hz
   4. 2500 Hz
5. If the second harmonic of an open pipe is 600 Hz, what is its fundamental frequency?
   1. 150 Hz
   2. 200 Hz
   3. 300 Hz
   4. 400 Hz
6. What is the wavelength of the fundamental mode in an open pipe of length 2m?
   1. 1 m
   2. 2 m
   3. 4 m
   4. 8 m
7. What is the wavelength of the fundamental mode in a closed pipe of length 2m?
   1. 2 m
   2. 4 m
   3. 6 m
   4. 8 m