Solved examples of Harmonic Frequencies in Closed Pipes

📚 Quick Study Guide

• 📏 Definition: A closed pipe is a pipe that is closed at one end and open at the other. Only certain frequencies of sound waves can resonate within the pipe.
• 🎵 Fundamental Frequency (1st Harmonic): The lowest frequency at which resonance occurs. The wavelength is four times the length of the pipe ($λ = 4L$). The frequency is given by $f_1 = \frac{v}{4L}$, where $v$ is the speed of sound and $L$ is the length of the pipe.
• 🎼 Harmonic Series: Closed pipes only support odd harmonics (1st, 3rd, 5th, etc.). Even harmonics are absent.
• 📊 General Formula for Frequency: The frequency of the $n$-th harmonic in a closed pipe is given by $f_n = n \frac{v}{4L}$, where $n = 1, 3, 5, ...$
• 🔊 Wavelengths: The allowed wavelengths are given by $λ_n = \frac{4L}{n}$, where $n = 1, 3, 5, ...$
• 💨 Speed of Sound: Usually considered as 343 m/s at room temperature (20°C), but can vary depending on temperature and the medium.

🧪 Practice Quiz

1. A closed pipe has a length of 0.5 meters. If the speed of sound is 343 m/s, what is the fundamental frequency?
   1. 85.75 Hz
   2. 171.5 Hz
   3. 343 Hz
   4. 686 Hz
2. What is the wavelength of the third harmonic in a closed pipe of length 1.2 meters?
   1. 4.8 m
   2. 1.6 m
   3. 0.8 m
   4. 0.4 m
3. A closed pipe resonates at frequencies of 400 Hz and 1200 Hz, with no intervening resonances. If the speed of sound is 340 m/s, what is the length of the pipe?
   1. 0.2125 m
   2. 0.425 m
   3. 0.6375 m
   4. 0.85 m
4. Which of the following harmonics is NOT possible in a closed pipe?
   1. 1st
   2. 3rd
   3. 4th
   4. 5th
5. The fifth harmonic in a closed pipe is 850 Hz. What is the fundamental frequency?
   1. 170 Hz
   2. 4250 Hz
   3. 85 Hz
   4. 50 Hz
6. If the length of a closed pipe is doubled, what happens to the fundamental frequency?
   1. It doubles
   2. It halves
   3. It remains the same
   4. It quadruples
7. What is the ratio of the frequency of the third harmonic to the fundamental frequency in a closed pipe?
   1. 1:3
   2. 3:1
   3. 1:2
   4. 2:1