OCR Specification focus:
‘Determine speed of sound using stationary waves in a resonance tube.’
Measuring the speed of sound using resonance links practical investigation with wave theory, demonstrating how stationary waves form in air columns and how wavelength and frequency reveal sound velocity.
Resonance and Stationary Waves in Air Columns
When sound waves are produced in a closed tube (a tube with one end sealed) and the other end open to the air, reflections of the wave at the closed end can interact with incoming waves. Under specific conditions, these reflected waves combine with the incident waves to form a stationary wave. Resonance occurs when the sound frequency leads to constructive interference between these waves, causing a significant increase in the amplitude of vibration in the air column.
At resonance, the tube length corresponds to a precise fraction of the sound wavelength, allowing the wavelength to be determined experimentally. The speed of sound can then be calculated using the relationship between frequency and wavelength.
Nature of Stationary Waves in a Resonance Tube
A stationary wave is formed when two waves of the same frequency and amplitude travel in opposite directions and interfere. In the case of sound, this happens between the source wave and its reflection.
Stationary Wave: A wave pattern formed by the superposition of two waves travelling in opposite directions with the same frequency and amplitude, resulting in regions of no displacement (nodes) and maximum displacement (antinodes)
In a closed–open tube:
The closed end is a node, where air cannot move.
The open end is an antinode, where air particles vibrate with maximum amplitude.
Only specific wavelengths fit this condition, producing resonance at distinct lengths.
Conditions for Resonance in Closed–Open Tubes
For the first few resonant modes:
1st harmonic (fundamental): The tube length L1L_1L1 ≈ ¼ of the wavelength (λ/4).
3rd harmonic: The tube length L2L_2L2 ≈ ¾ of the wavelength (3λ/4).
5th harmonic: The tube length L3L_3L3 ≈ 5λ/4, and so on.
Thus, the distance between successive resonant positions corresponds to half the wavelength (λ/2). This regular spacing allows for accurate wavelength measurement by identifying consecutive resonant lengths.
Using Resonance to Measure the Speed of Sound
The experiment typically involves a resonance tube, a tuning fork (or signal generator with loudspeaker), and a graduated cylinder partially filled with water.
Experimental Setup
A vertical tube is placed in water, open at the top and closed by water at the bottom.
A tuning fork of known frequency (f) is struck and held above the open end.
Resonance-tube apparatus with a tuning fork above a vertical tube and an adjustable water reservoir forming the closed end. Varying the water level changes the effective air-column length to locate loud resonances — the standard method for determining λ and hence v = fλ. Source.
The air column length is adjusted by moving the tube or altering the water level until resonance is heard — a loud, clear sound indicates a resonant condition.
This process is repeated to identify successive resonance positions, allowing the wavelength of sound in air to be determined.
Relationship Between Measured Quantities
EQUATION
—-----------------------------------------------------------------
Wave Equation (v) = fλ
v = Speed of sound in air (m s⁻¹)
f = Frequency of the tuning fork (Hz)
λ = Wavelength of sound in air (m)
—-----------------------------------------------------------------
Using the measured distance between resonance positions (corresponding to λ/2), the wavelength is determined, and with known frequency, the speed of sound is calculated.
Figure 17.25 illustrates a tuning fork driving a tube closed at one end. The standing-wave displacement has a node at the water surface and an antinode at the open mouth, with L = λ/4, highlighting the fundamental resonance condition used in this experiment. Source.
For example, if the difference between the first and second resonant lengths is 0.17 m, the wavelength is 0.34 m. With a tuning fork of 1000 Hz, v = 1000 × 0.34 = 340 m s⁻¹ (though calculations are excluded here, this illustrates proportional reasoning).
The End Correction
In practice, the antinode of the stationary wave does not form exactly at the physical end of the tube but slightly beyond it. This displacement is accounted for by an end correction to improve accuracy.
End Correction: A small adjustment added to the effective length of the air column to account for the fact that the antinode occurs slightly outside the open end of the tube.
Typically, the end correction (e) is approximately 0.6 × tube radius, and must be included when calculating the total effective length (L + e) for accurate wavelength determination.
Practical Procedure
To determine the speed of sound experimentally:
Set up the apparatus with a vertical tube partially immersed in water, ensuring the open end is above water.
Select a tuning fork of known frequency and strike it gently.
Hold the fork above the tube and slowly raise or lower the tube (or adjust the water level) until the loudest sound — resonance — is heard.
Measure the air column length at resonance using a metre ruler.
Repeat the process for multiple resonances to reduce uncertainty.
Calculate the wavelength, using the difference between consecutive resonant lengths to determine λ/2.
Compute the speed of sound using v = fλ.
These steps demonstrate the direct relationship between measurable physical quantities and the theoretical model of sound propagation.
Factors Affecting Accuracy
The accuracy of the speed of sound measurement depends on controlling and accounting for various factors:
Temperature: The speed of sound in air increases with temperature. Corrections should be made using known temperature dependence (approximately +0.6 m s⁻¹ per °C above 0 °C).
End correction: As noted, failure to include this leads to systematic underestimation of wavelength.
Frequency precision: The tuning fork frequency must be accurately known; any deviation affects the calculated speed.
Ambient conditions: Humidity and air pressure also influence sound speed slightly and should be recorded for completeness.
Measurement error: Parallax and reading inaccuracies when noting resonance lengths can affect precision.
Theoretical and Experimental Links
This experiment not only measures the speed of sound but reinforces the connection between standing wave theory and real-world acoustic behaviour. It demonstrates:
How boundary conditions (node at closed end, antinode at open end) determine resonant frequencies.
How wave behaviour in gases parallels that in strings and microwaves.
The practical method of deriving physical constants from wave phenomena.
Through resonance measurement, the quantitative link between frequency, wavelength, and wave speed — the foundation of wave physics — is directly observed in a simple yet powerful experiment.
FAQ
A closed–open tube naturally forms a node at the closed end and an antinode at the open end, creating the simplest possible standing wave pattern with the fewest variables.
Using a closed tube allows clear, distinct resonances at odd multiples of λ/4, which are easier to identify by ear.
Open–open tubes resonate at both even and odd harmonics, making it harder to determine which harmonic is being measured without additional equipment.
The speed of sound in air depends on the average kinetic energy of air molecules, which increases with temperature.
An approximate relationship is:
Speed increases by about 0.6 m s⁻¹ per °C above 0 °C.
This means measurements taken on a cold day will give a smaller value for sound speed unless a correction for temperature is applied. Controlling or recording the room temperature ensures more accurate, reproducible results.
The sound waves may not enter the air column efficiently, leading to weak or unclear resonances.
To ensure maximum energy transfer:
Hold the tuning fork vertically above the open end, with the prongs vibrating horizontally.
Keep the fork close to, but not touching, the tube opening.
Incorrect orientation can cause misalignment of the pressure waves, resulting in inaccurate resonance length readings and inconsistent results.
The boundary conditions restrict the wave pattern. A closed end must always be a node, and the open end must always be an antinode.
This configuration only allows standing waves that fit one-quarter, three-quarters, five-quarters of a wavelength, and so on.
Even harmonics (like λ/2 or λ) would require two antinodes or two nodes, which cannot exist simultaneously in a tube with one end closed and one end open.
Several techniques improve measurement reliability:
Use a finely graduated ruler to measure air column length precisely.
Repeat the experiment for several tuning fork frequencies and plot a graph of wavelength versus 1/frequency to reduce random error.
Mark multiple resonance points and take the mean of their spacing for a better estimate of λ/2.
Ensure quiet surroundings, as background noise can mask or distort the resonant sound.
These steps collectively reduce both systematic and random uncertainties in the determination of sound speed.
Practice Questions
Question 1 (2 marks)
A student uses a resonance tube closed at one end to determine the speed of sound in air. The first resonance occurs when the air column length is 0.18 m and the second resonance occurs at 0.52 m. The tuning fork frequency is 500 Hz.
(a) Calculate the speed of sound in air using these results.
Mark Scheme:
Step 1: Recognise that the difference between successive resonances corresponds to half a wavelength (λ/2).
(1 mark) λ = 2 × (0.52 − 0.18) = 0.68 m
Step 2: Apply the wave equation v = fλ.
(1 mark) v = 500 × 0.68 = 340 m s⁻¹
Question 2 (5 marks)
In a laboratory experiment, a student investigates resonance in a tube closed at one end using a tuning fork of known frequency.
(a) Explain how stationary waves are formed in the air column and describe how the student can determine the speed of sound in air using this method.
(b) State and explain one factor that must be taken into account to improve the accuracy of the result.
Mark Scheme:
(a)
Stationary waves form due to superposition of the incident and reflected sound waves of the same frequency travelling in opposite directions. (1 mark)
The closed end is a node, and the open end is an antinode. (1 mark)
Resonance occurs when the air column length is a quarter of the wavelength (λ/4) or an odd multiple of this. (1 mark)
Measure the distance between successive resonances to find λ/2, then use v = fλ to calculate the speed of sound. (1 mark)
(b)
Apply an end correction because the antinode occurs slightly outside the open end, affecting the measured length. (1 mark)
Or: Account for temperature, as the speed of sound in air increases with temperature. (1 mark, alternative)
