Open-Ended Resonance Tube Set

Introduction

Slide that trombone—change the length of the column of resonating air and the sound changes. Do it skillfully and the resonating noise is turned into music!

Concepts

• Resonance
• Harmonics
• Node
• Antinode
• Sine wave
• Wavelength

Background

Many musical instruments work because air is vibrated in an air column and then the length of the air column is varied to change the sound produced. The length of the air column determines the pitch of the sound of the vibrating air. A mixture of different frequencies and the resonation of air columns on a particular set of frequencies can turn noise into music. Changing the length of the column of vibrating air can vary the pitch of the instrument. The sound produced is the loudest when the air column is in resonance (in tune) with the vibrational source.

How does resonance occur? A vibrating source produces a sound wave. This wave of alternating high- and low-pressure variations moves through the air column. The sound wave is ultimately reflected back toward the vibrational source. It is either reflected back off a closed end of the column or as a low-pressure reflection off the open end of the column. If the reflected wave reaches the vibrational source at the same moment another wave is produced, then the leaving and returning waves reinforce each other. This reinforcement, called resonance, is achieved and a standing wave is produced. A standing wave is a pattern that results when two waves of the same frequency, wavelength, and amplitude travel in opposite directions and interfere with each other. A node is a point in a standing wave that always undergoes complete destructive interference and therefore is stationary. An antinode is a point in the standing wave, half-way between two nodes, at which the largest amplitude occurs (see Figure 1).

Standing wave patterns are often set up in musical instruments that are plucked, or bowed. They are also set up in wind instruments by the vibrations of a reed or musicians’ lips. Standing wave patterns are only created at an instruments’ natural frequencies, also known as harmonic frequencies. The harmonic series for a string instrument and an open-ended air column are shown in Figure 2. Notice that for a string instrument, a node is present at the start and end of the standing wave pattern. This is because the ends of the strings are fixed and not allowed to vibrate. This is not true for wind instruments containing an open-ended air column. In an open-ended air column, the sound wave traveling through the tube is reflected back on itself by the air molecules outside of the tube. When the reflected wave interferes with an incident wave, an anti-node is present at the start and end of the standing wave pattern. Look again at Figure 2 which represents the harmonic series. Notice the formulas written for each harmonic. These formulas were derived from the common formula below, which represents how the speed (v), wavelength (λ) and frequency (ƒ) of a wave are related. The formulas can be used to calculate the frequency of each harmonic, as long as the speed of sound and the wavelength are known. Further analysis of Figure 2 shows that in order to calculate the wavelength (λ) for each harmonic of a string instrument, the length (L) of the string and number of waves must be known. The first harmonic shows ½ of a wave present within the string, therefore the string length must be doubled to get one complete wavelength. The second harmonic shows one complete wave within the string, therefore the string length and wavelength are equal. The harmonic pattern continues, as seen in Figures 2 and 3. By knowing the instrument’s string length and the number of waves, the actual wavelength can be calculated for each harmonic. The formulas in Figure 2 can also be used for calculations involving the open-ended air column. The only difference here is that you would need to know the column length (L) for each harmonic instead of the string length. Although the standing wave patterns of a string instrument look different than an open-ended air column, the number of waves in each harmonic is the same. For example, the 1st harmonic for the opened-ended column shows ½ of a wave within the air column (this may be hard to see). The air column length must be doubled to get one complete wavelength. This is exactly what was done for the string instrument in the first harmonic. Further analysis of the diagrams in Figure 2 reveal that the number of waves present in each harmonic is the same for both the string instrument and open-ended air column.

Many musical instruments also contained closed-end air columns. A clarinet is a good example of this. As you might have guessed, in a closed-end instrument, the column is covered at one end. If a sound wave is traveling through this column, it will eventually be reflected off the closed end. The closed end will actually cause the reflected wave to invert. Therefore in a closedended air column, the standing wave pattern is produced due to the interference of an inverted reflected wave and an incident wave. Because of this, the standing wave pattern formed in a closed-end column will be different than the open-end column. The reason is that the closed-end acts as a fixed point which prevents movement. Therefore the closed-end of the column will always contain a node. However, in the open side of this column, the air will be free to move therefore it will always contain an anti-node. This is why the first Harmonic in a closed-end air column only contains ¼ of a wave (see Figure 4). Notice that the closed-end air column in Figure 4 does not have a second harmonic. The second harmonic of an instrument should always be twice the frequency of the first harmonic. The closed-end column can not have a second harmonic because it must always have a node at one end and an anti-node at the other. Therefore if we add another node and anti-node to the first harmonic, you will get ¾ of a wave, which is the third harmonic. Further analysis will reveal that a closed-end column will only produce odd harmonics. Because of this, calculating the wavelength of sound in a closed-end column is different than in the open-end column (see Figures 4 and 5).

Materials

Safety Precautions

This activity is considered safe but normal laboratory safety rules should be followed. Be sensitive to anyone who might have a hearing problem. Tuning fork vibrations might bother some individuals. Never touch fragile objects (e.g., glass, teeth) with a vibrating tuning fork.

Disposal

All materials can be reused many times.

Procedure

Open-Ended Column

1. Work with a lab partner. One person should hold the tube apparatus while the other uses the tuning fork.
2. Slide the shorter cardboard tube inside the longer tube. Allow it to protrude from the tube enough to grasp the end.
3. Set the rubber pad on the table. Strike the tuning fork firmly on the rubber pad.
4. Quickly place the tuning fork near the end of the tube while sliding the tubes inside each other varying the total length of the open-ended column (see Figure 6).
   {12886_Procedure_Figure_6_Open-ended resonance setup}
5. Listen carefully and repeat the process until a tube position is located where the resonant sound is the loudest. (Note how varying the length affects the sound.)
6. Measure the length of the extended tube at the point with the greatest resonance (loudest sound).
7. Ignoring the end corrections, which are negligable, the simplest standing wave pattern in a tube open at both ends is onehalf the wavelength. The wavelength in air would then be twice the length of the tube. (The wavelength for this tube is two times the length of the tube at resonance plus an additional correction of 3.8 cm for the difference in the size of the two ends.) Complete any calculations as directed by the instructor.

Closed-Ended Column

1. Use only the large diameter tube for this experiment. Carefully and slowly insert the foam plug into one end of the larger diameter tube.
2. Use the short tube to push the plug exactly 7.5 cm from one end and 15.5 cm from the other. Remove the short tube (see Figure 7).
   {12886_Procedure_Figure_7_Closed-ended resonance setup}
3. Strike the tuning fork and hold it at the end of the 15.5 cm end of the plugged tube. It should resonate and sound like the open-ended tube sound.
4. Now strike the tuning fork again and hold it at the 7.5 cm end of the plugged tube. Note the difference in the sound.
5. Experiment with different distances with a plugged end. How many different “tones” can be achieved?
6. Use a ruler or other long object to remove the plug from the tube so that it is ready for the next open-ended experiment.

Teacher Tips

• The sliding resonance tube activities can be conducted as just “listening” labs or they can be quantified with calculations. Consider course goals and student population when contemplating the teaching strategy for this laboratory.
• Relate the results of this lab to actual musical instruments. Bring clarinets, saxophones or trumpets to class and relate column length to resonance.
• Calculations for velocity might resemble the following depending upon specific data:
  

  
  Close-Ended Column

  {12886_Tips_Equation_2}

  Open-Ended Column

  {12886_Tips_Equation_3}