Transcript
Resonance Air Columns Theory: A transverse wave is a wave whose particles vibrate at right angles to the direction the wave is travelling. A longitudinal wave is a wave whose particles vibrate along the direction the wave is travelling. A solid can have a transverse or a longitudinal wave travel through it. However, a fluid or a gas can have only longitudinal waves. A longitudinal wave is composed of regions called compressions (or condensations) and rarefactions. A compression is a region where the molecules are compressed together. Likewise, rarefaction is a region where the molecules are spread out. The distance between any two adjacent rarefactions is the wavelength. The velocity of a longitudinal wave depends only on the properties of the medium it is traveling through. For a gas, the velocity of a longitudinal wave is given by
γ RT , (1.1) ρ where γ= Cp/Cv (γ = 1.40 for Air), R is the universal gas constant (R = 8.314 J/Mole), M is the molecular weight (28.8 x 10-3 kg/mole for air) and T is temperature in Kelvin degrees. The velocity of sound in air depends only on the temperature in Kelvin. v=
Standing waves are created in a vibrating string when a wave is reflected from an end of a string so that the returning wave interferes with the original wave. Standing waves also occur when a sound wave is reflected from the end of the tube. Resonance occurs when at certain frequencies of oscillation or at certain length of tubes, where all the reflected waves are in phase resulting in a large amplitude standing wave. The relationship between the length of the tube and the frequencies at which resonance occur depend on whether the end of the tube is open or closed. For an open tube having a length of L, resonance occurs when the wavelength (λ) satisfies the condition
λn =
2L ,n = 1,2, 3, 4,... n
(1.2)
fn =
nv ,n = 1,2, 3, 4... 2L
(1.3)
The resonance frequencies are given by
where v is the speed of sound, from equation (1.1) above. Equipment: 1. Pasco Resonance Air tube 2. Function generator 3. Dual Channel 4. Microphone
Procedures: Open Organ Pipe 1. Set up the oscilloscope, function generator, microphone and resonance tube as shown by the instructor. The microphone is attached to probe rod and the probe rod is pulled out so the microphone is even with speaker. Leave the plunger out of the tube. The function generator should be plugged into channel 1 of the oscilloscope, and the microphone is connected to channel 2. Set the oscilloscope sweep speed to approximately 5 ms/div, the gain on channel one and two to 5 mV/div, the trigger to channel 1, and the display to dual trace. Turn on the oscilloscope, the frequency generator and microphone amplifier. Set the output of the frequency generator to approximately 100 Hz. Adjust the amplitude of the function generator until you can distinctly hear sound from the speaker. Warning: You can damage the speaker by overdriving it. The sound from the speaker should be clearly audible, but not loud. Note also that many signal generators become more efficient and thus produce a larger output as the frequency increases, so you may need to reduce the amplitude as you increase frequency. The first four resonance states are diagramed in Fig. 1 below. Since both ends are open, an antinode must exist on each end. The simplest standing wave is for antinodes on each end and a node in the middle. This first resonance state (n = 1) is called the fundamental. Successive resonance states (n = 2, 3 , 4, ....) are called overtones. Notice that the 2nd overtone is twice the frequency of the fundamental, 3rd overtone it three times the fundamental frequency, and so on.
For a tube closed on one end, the simplest standing wave is for an antinode at the open end and a node at the closed end. This is the fundamental frequency. The 1st overtone has an antinode at open end and a node in the middle and a node at the end. The first four resonance states are shown in Fig. 2 below. The wavelength (λ) is given by
and the resonance frequencies are given by
4L ,n = 1,2, 3, 4... 2n − 1
(1.4)
(2n − 1)v ,n = 1,2, 3, 4... 4L
(1.5)
λ=
fn =
Notice that the 2nd overtone frequency is three times the fundamental frequency and 3rd overtone frequency is five times the fundamental frequency and so on.
2. Record the room temperature. Calculate the velocity of sound in air using Equation (1.1) and show the calculation and the results in the data table. 3. Increase the frequency slowly and listen carefully. Channel 1 is the speaker output and its amplitude should remain constant. You can adjust the frequency of the function generator if it doesn't. Channel 2 is the output of the microphone and its amplitude should increase as the frequency approaches the resonance frequency, reach a maximum at the resonance frequency and decrease after the resonance frequency is pasted. You this method to find the first six resonance frequencies of an open organ pipe. Record the resonance frequencies in the Data table. Find the ratio of the resonance frequency to the fundamental frequency. 4. Mount the microphone on the end of the probe and insert it into the tube through the hole in the speaker/microphone stand. (If not already done.) Set the frequency for the 3rd overtone. As you move the microphone down the length of the tube, note the positions where the signal is maximum and minimum. The distance between any two adjacent maxima or minima is one half a wavelength. Calculate all the possible distances between adjacent maxima or adjacent minima and enter the results in the table. Average the values and use the average to find the wavelength. The speed of sound is the product of the frequency times the wavelength. Find a percent error. Closed Tube 5. Place the plunger in the tube so that the length of the tube is 70 cm or less. Record the length of the tube in the data table. The microphone is attached to probe rod and the probe rod is pulled out so the microphone is even with speaker. Adjust the frequency slowly and find the first six resonance frequencies as one done in procedure 2. 6. Set the frequency for the 3rd overtone. As you move the microphone down the length of the tube, note the positions where the signal is maximum and minimum. Repeat the calculation in procedure 4 to find the speed of sound.
Data: A. Calculation of Speed of Sound in air from the room temperature Room Temperature = ___________oC = ______________K
Velocity of Sound in Air (Eqn. 1.1) = _______________________ (Show calculation below)
B. Open tube Length of open tube = _________________ n Frequency Fundamental (f0) 1st Overtone (f1) nd 2 Overtone (f2) rd 3 Overtone (f3) th 4 Overtone (f4) th 5 Overtone (f5)
(fn/f0)
C. Maxima and Minima of an open tube and the velocity of sound Resonance frequency = ______________________ 1 2 3 4
Microphone Maxima
Positions Minima
λ /2
Average wavelength = ______________________ Average Velocity of Sound = _____________ Percent Error = _____________
D. Closed tube Length of closed tube = _________________ n Frequency (fn/f0) Fundamental (f0) 1st Overtone (f1) nd 2 Overtone (f2) rd 3 Overtone (f3) th 4 Overtone (f4) th Overtone (f ) 5 5 E. Maxima and Minima of a closed tube and the velocity of sound Resonance frequency = ______________________ 1 2 3 4
Microphone Maxima
Positions Minima
λ /2
Average wavelength = ______________________ Average Velocity of Sound = _____________ Percent Error = _____________ Questions: There are some holes in the side of the resonance tube that are similar to holes in musical instruments. If a hole is open to the atmosphere, how does it effect the resonance frequency? Try it! Use the probe to find out if the holes produce a maximum or a minimum.
