9.5 Experimental Setup and Procedure

A stationary wave (standing wave) forms in an air column when a sound wave traveling down the column reflects off a surface and interferes with the original incident wave. This phenomenon is effectively demonstrated using a resonance tube, where specific lengths of the air column resonate with a sound source, producing a loud, clear tone.

Experimental Setup and Procedure

A simple experiment to demonstrate stationary waves in an air column uses the following apparatus:

• A hollow glass tube (resonance tube).
• A reservoir of water connected by a flexible hose, allowing the water level in the tube to be adjusted.
• A tuning fork of a known frequency.

Procedure:

1. Strike the tuning fork to make it vibrate and hold it just above the open end of the glass tube.
2. Slowly lower the water level in the tube, which increases the length of the vibrating air column.
3. Listen for a significant increase in the loudness of the sound. This loud sound indicates that resonance has occurred.
4. Mark the water level for the first point of resonance.
5. Continue lowering the water level to find subsequent points of resonance, which will also produce a loud sound.

Resonance and Stationary Wave Formation

• Resonance occurs when the frequency of the tuning fork matches one of the natural frequencies of the air column. At resonance, energy is efficiently transferred from the tuning fork to the air column, causing the air to vibrate with a large amplitude, which we hear as a much louder sound.

• How stationary waves form: The incident wave from the tuning fork travels down the tube and reflects off the water surface. The superposition of this incident wave and the reflected wave (traveling in opposite directions with the same frequency and amplitude) produces a stationary wave with fixed nodes and antinodes.

• Boundary Conditions: The interference between the incoming wave from the tuning fork and the wave reflecting off the water surface creates a stationary wave with specific boundary conditions:

  • A displacement node (point of zero air molecule displacement) must form at the closed end (the water surface).
  • A displacement antinode (point of maximum air molecule displacement) forms at or near the open end of the tube.

Modes of Vibration (Harmonics)

For a tube closed at one end, resonance occurs only when the length of the air column ($L$) allows a node to form at the closed end and an antinode at the open end. This happens at specific lengths related to the wavelength ($\lambda$) of the sound.

• First Resonance (Fundamental Frequency / 1st Harmonic): The shortest air column that produces resonance has a length of one-quarter of a wavelength. $L_1=\frac{\lambda }{4}$

• Second Resonance (1st Overtone / 3rd Harmonic): The next resonance occurs when the air column is three-quarters of a wavelength long. $L_2=\frac{3\lambda }{4}$

• Third Resonance (2nd Overtone / 5th Harmonic): The next resonance occurs at five-quarters of a wavelength. $L_3=\frac{5\lambda }{4}$

Only odd multiples of $\lambda /4$ can produce stationary waves in a tube closed at one end. This means only odd harmonics ($f_1,3f_1,5f_1,\dots$) are present; even harmonics are absent.

Determining the Speed of Sound

This experiment can be used to measure the speed of sound:

1. Find the air column lengths at the first ($L_1$) and second ($L_2$) resonance positions.
2. The distance between these two positions equals half a wavelength: $L_2-L_1=\frac{\lambda }{2}$
3. Calculate the wavelength: $\lambda =2(L_2-L_1)$
4. Using the known frequency ($f$) of the tuning fork, apply the wave equation: $v=f\lambda$

Why is the Sound Louder at Resonance?

At resonance, the air column's natural frequency matches the driving frequency of the tuning fork. This leads to constructive interference and an efficient transfer of energy, causing the air molecules to vibrate with a much larger amplitude, which is perceived as a louder sound.

Summary

• Stationary waves are formed in an air column by the superposition of an incident sound wave and its reflection.
• Resonance occurs when the frequency of the sound source matches a natural frequency of the air column, resulting in a loud sound.
• For a tube closed at one end, a displacement node forms at the closed end (water surface) and a displacement antinode forms near the open end.
• Resonance in a closed-end tube occurs at lengths that are odd multiples of a quarter-wavelength ($L=\lambda /4,3\lambda /4,5\lambda /4,\dots$).