9.9 Resonance of an Air Column | Waves | Physics 11

Resonance is a phenomenon that occurs when a vibrating system is driven by an external force at a frequency that matches one of its own natural frequencies of vibration. In the context of sound, when an air column inside a tube is exposed to a sound wave of a matching frequency, the air column will begin to vibrate with a very large amplitude, producing a significantly louder sound.

Figure 9.1: Stationary wave in a closed pipe

Stationary Waves (Standing Waves)

Resonance in an air column is the result of the formation of stationary waves.

Formation: A stationary wave is created by the superposition (interference) of two waves of the same frequency and amplitude traveling in opposite directions. In an air column, this happens when the incoming sound wave from a source (like a tuning fork) travels down the tube and reflects off the end. The incident and reflected waves then interfere.
Nodes and Antinodes: A stationary wave does not appear to travel. Instead, it has points of minimum or zero vibration called nodes and points of maximum vibration called antinodes.
- Displacement Node: A point where the air molecules have zero displacement.
- Displacement Antinode: A point where the air molecules oscillate with maximum amplitude.

Types of Waves

Transverse Waves: The particles of the medium vibrate perpendicular to the direction of wave propagation (e.g., waves on a string).
Longitudinal Waves: The particles of the medium vibrate parallel to the direction of wave propagation. Sound waves in air are longitudinal waves, consisting of compressions and rarefactions.

Open vs. Closed Pipes

The type of stationary wave that can form depends on the boundary conditions of the pipe.

a) Closed Pipe (Closed at one end, open at the other)

Boundary Conditions:
- The closed end must be a displacement node (air molecules cannot move).
- The open end is approximately a displacement antinode (air molecules are free to move with maximum amplitude).
Harmonics: Because of these constraints, a closed pipe can only support stationary waves where the length of the pipe is an odd multiple of a quarter-wavelength ( $L = \frac{1}{4} λ, \frac{3}{4} λ, \frac{5}{4} λ, \dots$ ). This means a closed pipe can only produce the fundamental frequency and its odd harmonics.

b) Open Pipe (Open at both ends)

Boundary Conditions: Both open ends must be displacement antinodes.
Harmonics: An open pipe can support stationary waves where the length of the pipe is any integer multiple of a half-wavelength ( $L = \frac{1}{2} λ, λ, \frac{3}{2} λ, \dots$ ). This means an open pipe can produce the fundamental frequency and all of its harmonics (both even and odd).

Formulas for Resonant Frequencies

The natural frequencies ( $f_{n}$ ) at which a pipe will resonate are determined by its length ( $L$ ) and the speed of sound ( $v$ ). The speed of sound in air depends on factors such as temperature and the medium through which it travels.

Open Pipe: $f_{n} = n \frac{v}{2 L} (for n = 1, 2, 3, \dots)$ The first harmonic ( $n = 1$ ) is the fundamental frequency $f_{1} = \frac{v}{2 L}$ .
Closed Pipe: $f_{n} = n \frac{v}{4 L} (for n = 1, 3, 5, \dots [odd integers only])$ The first harmonic ( $n = 1$ ) is the fundamental frequency $f_{1} = \frac{v}{4 L}$ .

End Correction

In practical experiments, the antinode at the open end does not form exactly at the tube's mouth but slightly outside it. This is known as end correction ( $e \approx 0.6 r$ , where $r$ is the inner radius of the pipe). The effective length of the pipe is therefore slightly greater than its physical length.

Possible Questions and Answers

Q: What is the difference between a harmonic and an overtone? A: Harmonics are all integer multiples of the fundamental frequency ( $f_{1}, 2 f_{1}, 3 f_{1}, \dots$ ). Overtones are all the frequencies produced by an instrument other than the fundamental frequency. In an open pipe, the first overtone is the second harmonic, the second overtone is the third harmonic, and so on. In a closed pipe, the first overtone is the third harmonic, the second overtone is the fifth harmonic, etc.
Q: How do wind instruments like a flute or a clarinet produce different musical notes? A: These instruments are essentially air columns. The musician changes the effective length of the air column by opening and closing holes along the instrument's body. A shorter air column produces a higher fundamental frequency (a higher pitch note), and a longer air column produces a lower frequency (a lower pitch note).

Stationary Waves (Standing Waves)

Resonance in an air column is the result of the formation of stationary waves.

Formation: A stationary wave is created by the superposition (interference) of two waves of the same frequency and amplitude traveling in opposite directions. In an air column, this happens when the incoming sound wave from a source (like a tuning fork) travels down the tube and reflects off the end. The incident and reflected waves then interfere.

Nodes and Antinodes: A stationary wave does not appear to travel. Instead, it has points of minimum or zero vibration called nodes and points of maximum vibration called antinodes.

Displacement Node: A point where the air molecules have zero displacement.
Displacement Antinode: A point where the air molecules oscillate with maximum amplitude.

Types of Waves

Transverse Waves: The particles of the medium vibrate perpendicular to the direction of wave propagation (e.g., waves on a string).

Longitudinal Waves: The particles of the medium vibrate parallel to the direction of wave propagation. Sound waves in air are longitudinal waves, consisting of compressions and rarefactions.

Open vs. Closed Pipes

The type of stationary wave that can form depends on the boundary conditions of the pipe.

Boundary Conditions:

The closed end must be a displacement node (air molecules cannot move).
The open end is approximately a displacement antinode (air molecules are free to move with maximum amplitude).

Harmonics: Because of these constraints, a closed pipe can only support stationary waves where the length of the pipe is an odd multiple of a quarter-wavelength (

L = \frac{1}{4} λ, \frac{3}{4} λ, \frac{5}{4} λ, \dots

). This means a closed pipe can only produce the fundamental frequency and its odd harmonics.

Boundary Conditions: Both open ends must be displacement antinodes.

Harmonics: An open pipe can support stationary waves where the length of the pipe is any integer multiple of a half-wavelength (

L = \frac{1}{2} λ, λ, \frac{3}{2} λ, \dots

). This means an open pipe can produce the fundamental frequency and all of its harmonics (both even and odd).

Formulas for Resonant Frequencies

The natural frequencies (

f_{n}

) at which a pipe will resonate are determined by its length (

L

) and the speed of sound (

v

). The speed of sound in air depends on factors such as temperature and the medium through which it travels.

Open Pipe: $f_{n} = n \frac{v}{2 L} (for n = 1, 2, 3, \dots)$ The first harmonic ( $n = 1$ ) is the fundamental frequency $f_{1} = \frac{v}{2 L}$ .

Closed Pipe: $f_{n} = n \frac{v}{4 L} (for n = 1, 3, 5, \dots [odd integers only])$ The first harmonic ( $n = 1$ ) is the fundamental frequency $f_{1} = \frac{v}{4 L}$ .

In practical experiments, the antinode at the open end does not form exactly at the tube's mouth but slightly outside it. This is known as end correction (

e \approx 0.6 r

, where

r

is the inner radius of the pipe). The effective length of the pipe is therefore slightly greater than its physical length.

Possible Questions and Answers

Q: What is the difference between a harmonic and an overtone? A: Harmonics are all integer multiples of the fundamental frequency ( $f_{1}, 2 f_{1}, 3 f_{1}, \dots$ ). Overtones are all the frequencies produced by an instrument other than the fundamental frequency. In an open pipe, the first overtone is the second harmonic, the second overtone is the third harmonic, and so on. In a closed pipe, the first overtone is the third harmonic, the second overtone is the fifth harmonic, etc.

Q: How do wind instruments like a flute or a clarinet produce different musical notes? A: These instruments are essentially air columns. The musician changes the effective length of the air column by opening and closing holes along the instrument's body. A shorter air column produces a higher fundamental frequency (a higher pitch note), and a longer air column produces a lower frequency (a lower pitch note).