Chapter 11 Wave Phenomena Name: Lab Partner: Section: 11.1 Purpose Wave phenomena using sound waves will be explored in this experiment. Standing waves and beats will be examined. The speed of sound will be determined. 11.2 Introduction While we will be concerned primarily with sound waves in this experiment, all waves have certain common features. There are two general types of waves. A transverse wave is one in which the disturbance is perpendicular to the direction of travel. A longitudinal wave is one in which the disturbance is parallel to the line of travel of the wave. Sound is a longitudinal wave. A wave is periodic if the pattern is repeated over and over. This occurs in both space and time. There are several features which describe such a periodic wave. The period, T, is thetime it takes foronecomplete cycle oroscillation ofthewave. The unitsareseconds. The wavelength, λ, is the distance between crest of the wave. Wavelength has units of distance. The amplitude is the magnitude of the disturbance. The amplitude can have various units for different types of waves. For a water wave, the amplitude would be the height in meters Transverse Longitudinal Direction of wave Figure 11.1: Transverse and longitudinal waves. Sound is a longitudinal wave where the displacement of the medium is in the same direction as the motion of the wave. 73
λ Velocity Amplitude Distance Figure 11.2: The amplitude and wavelength are represented in this figure. of the wave. For a sound wave, the amplitude is given in units of pressure (Pascal (Pa) = Newton/m 2 ) The period is related to the frequency by: f = 1 T (11.1) The unit of frequency is the Hertz (Hz = s 1 ). The relationship between the speed of a wave, the wavelength and the frequency is: v = λf = λ T (11.2) This applies to both transverse and longitudinal waves. For sound in air at 1 atmosphere of pressure (1.01 x 10 5 Pa) and 20 o C temperature, the velocity is 343 m/s Waves exhibit interference effects. This is based on the principle of superposition which can be stated as: When two or more waves are present simultaneously at the same place, the resultant wave is the sum of the individual waves. If the wavelength and amplitude of the two waves are the same, the two waves can add up in constructive interference to produce a wave with twice the amplitude of the individual waves (constructive interference). If the waves are out of phase by 180 o, i.e. one wave is going up while the other wave is going down, the result is destructive interference and the two waves cancel out. See figure 11.3. A standing wave is a stationary pattern produced by interference of a wave with itself when it reflects off of the opposite end of an enclosure. The wave interferes with its own reflection. Each standing wave pattern is produced at a unique frequency. This frequency corresponds to an integer number of wavelengths that will fit into the length, D, of the enclosure. The positions where there is no amplitude (pressure) is called a node. The positions with maximum amplitude (pressure) are called anti-nodes. Consider a tube of length D. The frequencies that produce the one loop, two loop, etc., patterns shown in figure 11.4 are given by: 74
+ = + = Figure 11.3: In the top schematic, the two wave interfere constructively. In the bottom schematic, the waves interfere destructively Figure 11.4: Standing wave pattern. f n = n( v ) n = 1,2,3,... (11.3) 2D where v is the velocity of the wave and D is the length of the tube. For a given D, a set of frequencies f n, where n = 1, 2, 3,..., produces standing waves. These frequencies are said to be harmonically related. The lowest frequency f 1 (n =1), is usually called the fundamental frequency, the next lowest frequency, f 2 = 2f 1 (n = 2), is called the first harmonic. When two waves of different frequency are added together, the two waves also interfere with one another. The number of times per second that the loudness rises and falls is the beat frequency. Consider two waves of the form: z(t) 1 = Asin(2πf 1 t) z(t) 2 = Asin(2πf 2 t) (11.4) Using the trigonometry identity sinα + sinβ = 2 cos 1(α β) 2 sin1 (α+β) the sum of the 2 two waves is: z(t) = 2Acos(( (2π(f 1 f 2 )t 2 ))sin(( 2π(f 1 +f 2 )t )) (11.5) 2 Note the difference of frequencies in the cosine function and the sum of the frequencies in the sine function. Figure 11.5 show a plot of a function like equation 11.5. The three variations in intensity form an envelope which is determined by the cosine function in equation 11.5.. 75
Figure 11.5: This figure shows a plot of a function like equation 11.5. Note the envelope which is determined by the cosine function in in equation 11.5. Transmitter Closed Tube L Reciever slide D Figure 11.6: Schematic diagram of the sound tube used to study standing waves and determine the speed of sound. 11.3 Procedure 11.3.1 Standing Waves A schematic of the closed sound tube is shown in figure 11.6. A photograph is shown in figure 11.7. When a continuous sound wave is generated by the transmitter (speaker) at one end of the tube, it reflects from the other (closed) end of the tube. The reflected wave interferes with later waves being generated by the transmitter. A standing wave pattern is set up in the tube. The movable receiver (microphone) is then used to detect the nodes and anti-nodes of the standing wave pattern. Part 1: Fixed distance with the frequency varied. Open the file wave1.ds in DataStudio. A signal generator and oscilloscope will appear. Pull the receiver to the end of the tube opposite the transmitter. Vary the frequency of the signal generator from 2 khz to 400 Hz. Record the four frequencies (f 1,f 2,f 3,f 4 ) from the signal generator where the amplitude is a maximum. The high frequency resonances (harmonics) are easier to see on the display so start at 2 khz and step the frequency down. You will find f 4 first, then f 3, etc. Verify the frequencies are related by equation 11.3, i.e. f 1 =f 4 /4, f 1 =f 3 /3, etc. Find the average f 1 (fundamental harmonic). 76
Figure 11.7: The closed tube apparatus is shown in this photograph. The closed sound tube speaker is connected to the signal generator output and the voltage input is connected to the sliding microphone. Harmonic Frequency f 4 f 3 f 2 f 1 average f 1 For the fundamental harmonic, f 1, the wavelength is 2 D where D is the length of the tube. Using the average fundamental harmonic, f 1, find the speed of sound from equation 11.2. Compare the result (percentage error) with the standard value 343 m/s. Speed of sound (v) Percent error Part 2: Fixed frequency with the distance varied. Find the f 4 harmonic around 1.8 khz. Record the frequency from the DataStudio signal generator display. Frequency (f 4 ) Place the receiver near the transmitter ( 2 cm). Slowly move the receiver across the tube and record the maxima (anti-nodes) in the table. The difference between maxima is L = λ 2 for f 4. Determine the average λ. Determine the speed of sound from equation 11.2 and compare to the standard value for the speed of sound (percent error). Average wavelength (λ) Speed of sound (v) Percent error 77
Maximum position (L) difference ( L) λ 11.3.2 Beats Figure 11.8: The experimental setup for measuring beat frequencies is shown in this photograph. Figure 11.8 shows the setup to measure the beat frequencies of two tuning forks with slightly different frequencies. Masking tape on one fork is used to slightly retard the frequency. Open the file wave2.ds. Adjust the sound detector to 1 cm from the forks. One fork should have masking tape to retard its frequency from the standard 256 Hz. Remove the tuning fork with the masking tape. Strike the tuning fork without the masking tape and click start. Expand the scale if necessary. Fit the curve with a sine fit and determine the frequency. The fit parameter for the sine fit is the period, T. Calculate the frequency from this period and record it below. Frequency of fork without tape Repeat for the other tuning fork with the masking tape and record the frequency below. Frequency of fork with tape 78
With both tuning forks in position, strike both tuning forks equally hard with the rubber mallet and click start. Beats should appear on the display i.e. a variation in the intensity like figure 11.5. Using the smart tool function in DataStudio, measure and record the time for four adjacent maxima of the beats. Calculate the time difference (period) between adjacent maxima and the frequency for these periods. Record the values in the table and calculate the average frequency. Maximum time difference (T) frequency Average frequency The frequency of the beats is the difference in the frequency between tuning forks with and without masking tape. Calculate the difference in frequency between the two tuning forks. Calculate the percentage difference between this difference in frequency and the average frequency of the beats calculated in the previous step. Difference in frequency between forks Percent difference Expand the scale so 50 cycles are visible near one of the maxima. Using a sine fit, determine the frequency. Compare (percentage difference) the sine fit frequency to one half of the sum of the two individual tuning fork frequencies. 11.3.3 Questions Frequency from sine fit Percent difference 1. Why does adding masking tape to one of the tuning forks lower the frequency? 79
2. Why is the frequency of the beats the difference in the two individual fork frequencies and not one half of this difference? 11.4 Conclusion Write a detailed conclusion about what you have learned. Include all relevant numbers you have measured with errors. Sources of error should also be included. 80