Vibration in Air Columns (HSC SSCE Physics): Revision Notes

Vibration in Air Columns

Introduction to air column vibration

Standing sound waves form in pipes through resonance. These waves can occur in two types of pipes:

• Open pipes: pipes that are open at both ends
• Closed pipes: pipes that are open at only one end (closed at the other)

Resonance happens when the wavelength of a sound wave corresponds exactly to one of the natural frequencies of the pipe. The resonance pattern depends on the pipe's length and the speed of sound in air, which varies with temperature.

Reflection of sound waves in pipes

Sound waves travel through pipes as plane waves, just as they do in open air. Understanding how these waves reflect at the ends of pipes is crucial to understanding resonance.

Reflection at open ends

When a compression travels along a pipe and reaches an open end:

• The compression is no longer confined
• It expands rapidly into the surrounding air
• This expansion creates a rarefaction that reflects back down the pipe
• Result: compression reflects as a rarefaction (similar to fixed-end reflection in strings)

When a rarefaction reaches an open end:

• Higher pressure air from outside the pipe rushes into the low-pressure region
• This creates a compression that travels back down the pipe
• Result: rarefaction reflects as a compression

Reflection at closed ends

At a closed end, waves reflect differently:

• A compression reflects as a compression (similar to free-end reflection in strings)
• A rarefaction reflects as a rarefaction
• The closed end creates a boundary condition for the standing wave

Nodes and antinodes in pipes

The standing wave pattern creates specific points in the pipe:

• Displacement antinode: maximum air displacement (occurs at open ends)
• Displacement node: zero air displacement (occurs at closed ends)
• Pressure node: minimum pressure variation (occurs at open ends)
• Pressure antinode: maximum pressure variation (occurs at closed ends)

Stationary waves in open pipes

In pipes open at both ends, standing waves form with specific patterns. We can describe these patterns using either particle displacement ($\Delta x$) or pressure variation ($\Delta p$).

Harmonics in open pipes

Open pipes support all harmonics. This means every integer multiple of the fundamental frequency can resonate in the pipe.

The relationship between pipe length ($l$) and wavelength ($\lambda_n$) is:

$l = n\frac{\lambda}{2}$

where:

• $l$ = length of the pipe
• $\lambda_n$ = wavelength of the $n$th harmonic
• $n$ = harmonic number ($n = 1, 2, 3, 4...$)

Frequency calculations for open pipes

The fundamental frequency (first harmonic) is:

$f_1 = \frac{v}{2l}$

where:

• $f_1$ = fundamental frequency
• $v$ = speed of sound in air
• $l$ = length of the pipe

Higher harmonics follow the pattern:

• 2nd harmonic (1st overtone): $f_2 = 2f_1$
• 3rd harmonic (2nd overtone): $f_3 = 3f_1$
• 4th harmonic (3rd overtone): $f_4 = 4f_1$

In general: $f_n = nf_1$ for all positive integers $n$.

Worked example: Open pipe length calculation

Stationary waves in closed pipes

Pipes closed at one end have different resonance characteristics than open pipes. This affects the sounds they can produce.

Harmonics in closed pipes

Closed pipes support only odd harmonics. This is a crucial difference from open pipes.

The relationship between pipe length and wavelength is:

$l = (2n-1)\frac{\lambda}{4}$

where:

• $l$ = length of the pipe
• $\lambda$ = wavelength
• $n$ = positive integer ($n = 1, 2, 3...$)

The term $(2n-1)$ always produces an odd number:

• When $n = 1$: $(2 \times 1 - 1) = 1$ (1st harmonic)
• When $n = 2$: $(2 \times 2 - 1) = 3$ (3rd harmonic)
• When $n = 3$: $(2 \times 3 - 1) = 5$ (5th harmonic)

Frequency calculations for closed pipes

The fundamental frequency for a closed pipe is:

$f_1 = \frac{v}{4l}$

Notice this is half the fundamental frequency of an open pipe of the same length.

The overtones are:

• 1st overtone (3rd harmonic): $f_2 = 3f_1$
• 2nd overtone (5th harmonic): $f_3 = 5f_1$
• 3rd overtone (7th harmonic): $f_4 = 7f_1$

In general: $f_n = (2n-1)f_1$ for odd harmonics only.

Musical implications

The fact that closed pipes only produce odd harmonics has an important consequence: musical instruments with closed pipes can only play certain notes. They produce a different timbre (sound quality) compared to open pipe instruments, as they're missing all the even harmonics.

Worked example: Comparing open and closed pipes

Investigation: Finding the speed of sound by air column resonance

This investigation uses resonance in a closed pipe to experimentally determine the speed of sound in air.

Aim

To find the speed of sound in air in your classroom.

Materials

• Large graduated cylinder
• Glass tube (approximately $2-3 \text{ cm}$ in diameter and about $30 \text{ cm}$ long)
• Tuning fork with a fundamental frequency of $440 \text{ Hz}$
• Ruler
• Marking pen that can write on wet glass

Risk assessment

What are the risks in doing this investigation?	How can you manage these risks to stay safe?
The glass tube may break and produce sharp edges	Take care when handling the glass equipment

Method

1. Set up the equipment as shown in the diagram. The glass tube fits inside the graduated cylinder and can be moved up and down to vary the length of the air column inside.
2. Fill the graduated cylinder with water to just below the top.
3. Lower the glass tube until the top is just above the surface of the water.
4. Strike the tuning fork on a surface that will get it oscillating strongly and hold it near the open end of the tube.
5. Raise the tube slowly until it begins to resonate. You will hear the sound become noticeably louder.
6. Using the marking pen, mark the base of the tube where it contacts the water when the loudest resonance is detected.
7. Measure the length of the air column from the top of the tube to the water level. This is the resonance length.
8. Repeat the procedure twice more and calculate the average of the three measurements.
9. Analyse the results using the relationship for the fundamental frequency in a closed pipe: $l = \frac{\lambda}{4}$.

Results

Record your results in the table below:

Data	Trial 1	Trial 2	Trial 3	Average
Frequency of the tuning fork
First resonance length

Analysis of results

1. Calculate the speed of sound using your data:
   • Since $l = \frac{\lambda}{4}$, the wavelength is $\lambda = 4l$
   • Using $v = f\lambda$, calculate: $v = f \times 4l$
   • Include the uncertainty in your calculated value
2. Compare your result with the accepted value of $340 \text{ m s}^{-1}$:
   • Calculate the percentage difference between your value and the accepted value
   • Percentage difference $= \frac{|\text{your value} - \text{accepted value}|}{\text{accepted value}} \times 100\%$

Discussion

1. Atmospheric conditions: What atmospheric conditions (temperature, humidity, air pressure) may have affected your result? The speed of sound increases with temperature, so if the room was warmer than standard conditions, your measured speed would be higher.
2. Temperature effects: If you conducted the experiment on a hotter day, the speed of sound would be greater. This would result in a longer resonance length for the same frequency, as $l = \frac{v}{4f}$ shows that length is directly proportional to the speed of sound.
3. Improving accuracy: Changes to improve accuracy could include:
   • Using a longer tube to reduce percentage error in measurements
   • Taking more trials and averaging them
   • Using a more precise method to measure the tube length
   • Controlling room temperature more carefully
4. Different gases: If you replaced the air with carbon dioxide, the speed of sound would change. Sound travels at different speeds in different media depending on the medium's properties (density and elasticity). Carbon dioxide is denser than air, so sound would travel more slowly in it.
5. Pressure diagram: When resonating at the fundamental frequency, there is a displacement antinode (pressure node) at the open top of the tube and a displacement node (pressure antinode) at the closed bottom where the water surface is.
6. Higher frequency tuning fork: If you used a tuning fork with a higher fundamental frequency, the first resonance would occur in a shorter air column. This is because the relationship $l = \frac{v}{4f}$ shows that length is inversely proportional to frequency - as frequency increases, the required resonance length decreases.

Conclusion

Write a conclusion that:

• States your calculated value for the speed of sound
• Compares it to the accepted value
• Discusses the accuracy of your result
• Identifies the main sources of error
• Relates your findings back to the aim of the investigation

Exam tips

Remember!