PC1141 Physics I. Speed of Sound. Traveling waves of speed v, frequency f and wavelength λ are described by

Transcription

1 PC1141 Physics I Speed of Sound 1 Objectives Determination of several frequencies of the signal generator at which resonance occur in the closed and open resonance tube respectively. Determination of the speed of sound from the linear least square fits to the data. Determination of the positions of node and antinode in the resonance closed tube for sixth harmonic (n = 7). Determination of the speed of sound from the measured positions of node and antinode. 2 Equipment List Resonance tube Signal generator Oscilloscope Vernier caliper 3 Theory Traveling waves of speed v, frequency f and wavelength λ are described by v = fλ (1) We can determine the speed of a traveling wave for known frequency and wavelength from equation (1). There are various indirect methods of measuring the wavelength λ, such as interference, diffraction, etc. However, it is difficult to measure the wavelength of a traveling wave directly. When two waves of exactly the same speed, frequency and wavelength travel in opposite directions in the same region, they produce standing waves. The wavelength of a standing wave can be easily determined by locating the nodes (the points of no vibration) and antinodes (the points of maximum vibration) and together with the known frequency of vibration, the speed of wave can be calculated. Page 1 of 5

2 Speed of Sound Page 2 of 5 Figure 1: Equipment setup for the resonance tube. In this experiment, a standing sound wave will be produced in a resonance tube (see Figure 1). The resonance tube consists of two transducers which can both act as sources or detectors of sound waves, rather like speakers and microphones of a sound system. The source transducer is connected to a signal generator which produces electrical signals at desired frequency and thus set to vibrations. The vibrations propagate down the length of the resonance tube but are reflected from the other end. A standing wave is then set up due to the superposition of the two waves traveling in opposite directions. The detector transducer is connected to an oscilloscope and is used to measure the amplitude of the standing wave at various locations inside the resonance tube. When the diaphragm of a speaker vibrates, a sound wave is produced that propagates through the air. The sound wave consists of small motions of the air molecules toward and away from the speaker. If you were able to look at a small volume of air near the speaker, you would find that the volume of air does not move far, but rather it vibrates toward and away from the speaker at the frequency of the speaker vibrations. This motion is analogous to waves propagating on a string. An important difference is that, if you watch a small portion of the string, its vibrational motion is transverse to the direction of propagation of the wave on the string. The motion of a small volume of air in a sound wave is parallel to the direction of propagation of the wave. Because of this, the sound wave is called a longitudinal wave. A standing sound wave has displacement nodes points where the air does not vibrate and displacement antinodes points where the amplitude of the air vibration is a maximum. However, the detector transducer does not sense the displacement of the air, it senses the pressure of the air. The oscilloscope then shows where pressure nodes and antinodes are occurring. It turns out that pressure nodes occur at displacement antinodes and pressure antinodes occur at displacement nodes. This can be understood by thinking of a pressure antinode as being located between two displacement antinodes that vibrate 180 out of phase with each other. When the air of the two displacement antinodes are moving toward each other, the pressure of the pressure antinode is a maximum. When they are moving apart, the pressure goes to a minimum. Reflection of the sound wave occurs at both open and closed tube ends. If the end of the tube is closed, the air has nowhere to go, so a displacement node (a pressure antinode) must exist at a closed end. If the end of the tube is open, the pressure stays very nearly at room temperature, so a pressure node (a displacement antinode) exists at an open end of the tube.

3 Speed of Sound Page 3 of 5 As described above, a standing wave occurs when a wave is reflected from the other end of the tube and the return wave interferes with the original wave. However, the sound wave will actually be reflected many times back and forth between the ends of the tube and all these multiple reflections will interfere together. In general, the multiply reflected waves will not all be in phase and the amplitude of the wave pattern will be small. However, at certain frequencies of oscillation, all the reflected waves are in phase, resulting in a very high amplitude standing wave. These frequencies are called resonant frequencies. Figure 2: Resonant states: Closed and Open Tubes. Depending on the frequency and the length of the air tube, various standing wave patterns can be set up when resonances occur (see Figure 2). For a closed tube (both ends are closed), resonance occurs when the wavelength of the wave satisfies the condition: L = n λ, n = 1, 2, 3, 4,... (2) 2 where L is the length of the air tube. These wavelengths allow a standing wave pattern such that a pressure antinode (displacement node) of the wave pattern exists naturally at each end of the tube. Another way to characterize the resonance states is to say that an integral number of half wavelengths fits between the ends of the tube. For an open tube (one end is open and the other end is closed), resonance occurs when the wavelength of the wave satisfies the condition: L = n λ, n = 1, 3, 5, 7, 9,... (3) 4 These wavelengths allow a standing wave pattern such that a pressure node (displacement antinode) occurs naturally at the open end of the tube and a pressure antinode (displacement node) occurs naturally at the closed end of the tube. As for the closed tube, each successive value of n describes a state in which one more half wavelength fits between the ends of the tube.

4 Speed of Sound Page 4 of 5 However, the above formula for the open tube is only approximate as the behaviour of the waves at the open end of the tube depends partially on factors such as the diameter of the tube and the frequency of the waves. The open end of the tube is not exact displacement antinode (pressure node). The node occurs just beyond the open end of the tube. The end correction is about 0.3 times the diameter of the tube. It then follows that resonance occurs when the wavelength of the wave satisfies the condition: L + 0.3d = n λ, n = 1, 3, 5, 7, 9,... (4) 4 The speed of sound in air depends slightly on the temperature of the air. For a limited range of temperatures, the dependence is approximately linear. If v T stands for the speed of sound at a temperature of T C, to an excellent approximation it is given by v T = ( T ) m/s (5) This equation will be used to determine the accepted value for the speed of sound in air. 4 Laboratory Work Part A: Resonant Frequencies of A Tube When a speaker vibrates near a tube, there are certain frequencies at which the tube will amplify the sound from the speaker. These frequencies are called resonant frequencies and occur because the dimensions of the tube are such that, at these frequencies, there occurs a maximum transfer of energy between the speaker and the tube. In this part of the experiment, you will investigate the relationship between the length of the tube and the frequencies at which resonance occurs. Both closed (both ends are closed) and open tube (one end is open and the other end is closed) will be studied. A-1. Determine the room temperature of the air and record it as T in Data Table 1. A-2. Setup the resonance tube, oscilloscope and signal generator as shown in Figure 1. A-3. Pull the microphone all the way back until its surface is flushed with the end plug. A-4. Measure the length of the air column and record it as L in Data Table 1. A-5. Turn on the function generator and set the output frequency of the function generator to approximately 100 Hz. Adjust the amplitude of the function generator until you can distinctly hear the sound from the speaker. A-6. Increase the frequency slowly and listen carefully. In general, the sound will become louder as you increase the frequency because the function generator and speaker are more efficient at higher frequencies. However, listen for a relative maximum in the sound level a frequency where there is a slight decrease in the sound level as you increase the frequency slightly. This relative maximum indicates a resonance mode in

5 Speed of Sound Page 5 of 5 the tube. Adjust the frequency carefully to find the lowest frequency at which a relative maximum occurs. (You can also find the relative maximum by watching the trace on the oscilloscope. When the signal is a maximum height, you have reached a resonant frequency.) Record the value of this lowest resonant frequency in Data Table 1. A-7. Increase the frequency slowly until a new resonant frequency is found. Again, measure and record the frequency in Data Table 1. A-8. Continue finding still higher resonant frequencies until SIX sets of data are obtained. Record these values of resonant frequency in Data Table 1. A-9. Now, remove the microphone and the supporting plug from the tube. Place the microphone close to the open end of the tube, being sure that the supporting plug is far from the opening. A-10. Measure the inner diameter of the tube and record it as d in Data Table 2. A-11. Repeat steps A-5 through A-8 for the open tube, recording the readings in Data Table 2. Part B: Standing Waves in A Tube A sound wave propagating down a tube is reflected back and forth from each end of the tube, and all the waves, the original and the reflections, interfere with each other. If the length of the tube and the wavelength of the sound wave are such that all of the waves that are moving in the same direction are in phase with each other, a standing wave pattern is formed. This is known as a resonance mode for the tube and the frequencies at which resonance occurs are called resonant frequencies. In this part of the experiment, you will set up standing waves inside the resonance tube (both closed and open tube) and use the microphone to determine the characteristics of the standing waves. B-1. With the tube closed, adjust the frequency of the signal generator to that of the sixth harmonic (n = 6). Record the value of this frequency in Data Table 3. B-2. Move the microphone from one end of the air tube to the other end and note the positions where the oscilloscope signal is a maximum and where it is a minimum. Determine FIVE positions respectively for both maximum and minimum. Record these positions in Data Table 3. B-3. With the tube open, adjust the frequency of the signal generator to that of the eleventh harmonic (n = 11). Record the value of this frequency in Data Table 4. B-4. Move the microphone from one end of the air tube to the other end and note the positions where the oscilloscope signal is a maximum and where it is a minimum. Determine FIVE positions respectively for both maximum and minimum. Record these positions in Data Table 4. Last updated: Monday 20 th October, :53pm (KHCM)