Speed of sound using a resonance tube

Practical method

Aim

Measure the speed of sound using resonance in a tube closed at one end, correcting for the end effect.

Variables

Independent variable: frequency of the tuning fork or speaker
Dependent variable: length of air column at first resonance
Control variables: air temperature; tube diameter.

Apparatus

• Resonance tube in a tall cylinder of water (or a tube with a movable piston)
• Tuning forks of known frequency or a small loudspeaker
• Metre rule
• Thermometer

Method

1. Hold a sounding tuning fork over the open end and raise the tube until the sound is loudest: this is resonance with the first harmonic.
2. Measure the air-column length L from the water surface to the top of the tube.
3. Repeat for several frequencies, approaching resonance from above and below.
4. The first resonance satisfies L + e = \(\dfrac{\lambda }{4}\), where e is the end correction.
5. Plot L against \(\dfrac{1}{f}\): gradient = \(\dfrac{v}{4}\) and the intercept on the L axis is −e.

Sample results

Air at 20 °C; tube diameter 2.0 cm.

Analysis

L against \(\dfrac{1}{f}\) is a straight line with gradient 85 m s⁻¹, so v = 4 × 85 = 340 m s⁻¹. The line meets the L axis at −0.006 m, so the end correction e = 0.006 m (about 0.3 × diameter).

Uncertainty

Locating the loudest resonance is uncertain by about ±3 mm in L (about 1–2%). The graph method removes the need to know e and averages the random errors.

Errors and improvements

Risk assessment

Common mistakes

• Taking L as half a wavelength (it is a quarter for a closed tube).
• Forgetting the end correction.
• Measuring L from the bottom of the cylinder.

Worked analysis

At 384 Hz, L = 0.215 m. With e = 0.006 m, \(\dfrac{\lambda }{4}\) = 0.221 m, so λ = 0.884 m and v = 384 × 0.884 = 339 m s⁻¹.

Common misconception

At first resonance a closed tube holds a quarter of a wavelength (node at the water, antinode just beyond the open end), not half.