PURPOSE The purpose of this experiment is to study the propagation of longitudinal waves through an air column, to measure the wavelength of these waves and hence determine the speed of sound in air. The measurement will be carried out by studying the resonance phenomenon that occurs in a simple organ pipe apparatus. THEORY The apparatus used in this experiment is pictured below in Figure 1. It consists of a long glass column partially filled with water. The level of water within the column can be adjusted by raising or lowering the reservoir mounted on the post next to the tube. The column of air between the level of the water and the open end of the tube forms an "organ pipe" closed at one end (the water surface) and open at the other. Let the length of this air column be denoted by L. We will use two tuning forks of different frequencies as the sources of sound. We know that the air column will resonate only for certain values of the length, L, of the column. Resonance occurs when standing waves are formed in such a way that a node is formed at the closed end (water surface) and an antinode is formed at (or near) the open end (top) of the tube. Then, exactly an odd number of quarter wavelengths fit within the length L. That is, L n = (2n-1)λ/4 n = 1, 2, 3, 4 (1) The sound waves undergo a phase reversal upon reflection at the water surface. Although sound waves are longitudinal waves they can be graphically represented as transverse waves for ease of depiction. The first three resonance conditions are shown schematically in Figure 2. We must remember that the waves are longitudinal as they propagate through the column. In this experiment we hold one of the vibrating tuning forks of frequency f 1, about 1-2 cm above the top edge of the tube. By adjusting the length L, we can obtain one of the resonance conditions shown in Figure 2. VI-1
The length is fine-tuned by listening for the maximum intensity of sound. The range of values of L for which no change in intensity can be detected defines the uncertainty in the length L. How do we know which resonance condition we have? That is, how do we determine the mode number 'n' in Equation 1? By starting with the water level very near the top of the tube and gradually lowering it, the first resonance condition we encounter will correspond to n = 1, with length L 1. The next will correspond to n = 2, having length L 2 and so on. For each of the three resonance conditions shown in Figure 2, the wavelength can be calculated from Equation 1. We get, 4L λ = n (2) 2 n 1 In practice, the resonance condition used to derive Equation 1 is only an approximation that becomes more accurate as the ratio of the tube diameter to the tube length decreases. The position at the open end where the antinode (maximum particle displacement) is formed is actually a finite distance L above the end of the tube. That is, the actual wavelength is larger than the value predicted by Equation 2. Figure 3 shows this situation schematically for the case n = 2. We will assume that L is constant for all resonance conditions (all n). (This assumption could always be checked by performing a more detailed experiment.) Therefore Equation 1 has to be corrected as For n=1 we have from Equation 3 2n 1 L n + L = λ, n=1, 2, 3 (3) 4 and for n = 2 we get L 1 λ + L = 4 L 3 3λ + L = 4 If we subtract the n = 1 resonance condition from the n = 2 resonance condition, L cancels and solving for the wavelength gives ( L ) λ = 2 L (4) 2 1 VI-2
A similar relation can be derived from the n = 3 and n = 1 resonance conditions. You are asked to carry out the calculation in your pre-lab. Look what has happened! We have solved the problem of determining the actual wavelength without having to make any more measurements. This is an example of how processing the same data with a different method can give more accurate results. The final step in computing the speed of sound is through the fundamental wave relation, v = fλ (5) We can calculate the average value of v using the average of the wavelength in equation (4) and the wavelength found from the difference between the n = 3 and n = 2 modes. Following the same procedure we can calculate the average value of speed of sound in air for the second tuning fork. Finally, these two numbers can be averaged to yield a single estimate for the speed of sound in air. Comparison with an alternate method: There is another, much simpler method for determining the speed of sound in air which requires just a single measurement. This measurement is the air temperature. You will learn later in thermodynamics (or may already know) that the speed of sound in an ideal gas is given by the following relation. γ R T v = M 1/ 2 (6) where T is the absolute temperature (degree Kelvin) of the gas. Here R is the gas constant whose value is R = 8.314 J mol -1 K -1 and M is the molecular mass of the gas. Since air is a composite gas, M is an average over the constituent gases. You are asked to compute M in the pre-lab taking air to be 78% N 2, 21% O 2 and 1% Ar. VI-3
Finally, γ is the ratio of specific heats (C p /C v ) of the gas. For an ideal diatomic gas at moderate temperatures (room temperature is moderate) γ = 7/5. Since air is essentially a mixture of diatomic gases, we will take this to be the value of γ. The laboratory thermometers measure the temperature, in degrees Celsius ( C), so you will need to convert to the Kelvin scale (K) using T(K) = T( C) + 273.15. The speed of sound is then calculated easily from Equation 6. This value can be compared with those obtained from the resonance method. EXPERIMENTAL PROCEDURE The apparatus for measuring the speed of sound is shown in Figure 1. It consists of a glass tube partly filled with air and partly filled with water. The tube is open at the top, and its length can be varied by changing the water level (by raising or lowering the reservoir) 1. Set the reservoir at its highest level and fill it with water until the water level is about 6 cm below the top of the tube. 2. Holding the tuning fork in your hand, strike the fork with the rubber mallet and hold it above the tube. Make sure that the fork vibrates along the vertical. DO NOT STRIKE THE FORK WITH ANYTHING BUT THE RUBBER MALLET. (Hard surfaces will damage the fork and change its frequency). DO NOT STRIKE THE FORK NEAR THE GLASS TUBE. (The glass will break if the fork hits it). 3. Lower the reservoir until resonance occurs (i.e. you hear maximum intensity of sound from the system). Measure the smallest column length L 1 that produces resonance. By raising and lowering the water level slightly, repeat your measurements at least three more times. Enter the frequencies (f 1 and f 2 ) of the tuning forks in Table 1. (Note: A spreadsheet template of the Table is provided in the software.) 4. Lower the reservoir until the second and third resonances are found. Record the column lengths (L 2 and L 3 and repeat your measurements) at least three more times in each case. 5. Repeat the steps 2, 3 and 4 for the second tuning fork and enter the values in Table 1. 6. For one resonant height, move the tuning fork to several different positions above the top end of the tube. This is to see whether the resonating length and the maximum intensity of sound heard change with the position of the fork. Record any changes in the resonant height. Measure the tube diameter. DATA ANALYSIS 1. Determine the mean value of L 1 (smallest length of air column that produces resonance) from your measurements for fork #1 and enter its value in Table 1. 2. Also determine the mean values of L 2 and L 3 and enter them in Table 1. VI-4
3. Use the mean values of L 1,L 2 and L 3 to compute the values of (L 2 - L 1 ) and (L 3 - L 1 ). Enter these in Table 1. 4. From these values of (L 2 - L 1 ) and (L 3 - L 1 ), find the values of λ 1 and v 1. 5. Repeat steps 1-4 for fork #2 and determine the values of λ 2 and v 2. 6. From the values of v 1 and v 2, find the average value of velocity of longitudinal waves in air. Enter this result in Table 1. 7. Measure the tube diameter. Estimate the value of L from average values of λ 1 and L 1 for fork #1. Repeat this for fork #2. Find the ratios of these values to the diameter of the tube. VI-5
NAME Sec/Group Date TABLE 1 Frequency of fork (1)...Hz (2)... Hz FORK LENGTH (cm) 1 2 3 Mean L n Difference (cm) 1 L 1 L 2 (L 2 - L 1 ) = L 3 (L 3 - L 1 ) = 2 L 1 L 2 (L 2 - L 1 ) = L 3 (L 3 - L 1 ) = Determination of average wavelength and velocity Fork 1: λ 1 /2 = (L 2 - L 1 ) λ 1 = 2(L 2 - L 1 ) =... cm λ 1 = (L 3 - L 1 ) =... cm Average λ 1 =... cm Velocity V 1 = f 1 λ 1 =... cm/s Fork 2: λ 2 /2 = (L 2 - L 1 ) λ 2 = 2(L 2 - L 2 ) =... cm λ 2 = (L 3 - L 1 ) λ 2 = (L 3 - L 1 ) =... cm Average λ 2 =...... cm Velocity V 2 = f 2 λ 2 =... cm/s Average value of velocity of longitudinal waves V = (V 1 + V 2 )/2 =... cm/s =... m/s Tube Diameter (d)... cm Fork 1 : Fork 2 : L = λ 1 /4 - L 1ave L = λ 2 /4 - L 1ave =... cm =... cm VI-6