Lab report 2

Report for Experiment #14 Standing Waves Abstract In this experiment, our main purpose is to study standing waves of a string and in an air column in order to test the wave velocity. These two are the completely related investigations. In investigation 1, we tested the standing waves of a string. We attached a plastic pail at one end of the string to created waves by adding weight (0.2968kg, 0.1678kg, 0.1034kg, 0.0738kg, 0.0603kg, 0.0501kg) and a 120Hz vibrator at another end of the string to show the number of Nodes(3,4,5,6,7,8). We then used weights to measure the tension(2.91N,1.64N,1.01N,0.72N,0.59N,0.49N) and the nodes number and length of the string(1.55m) to measure the velocity(124m/s, 93m/s,74.4m/s,62m/s, 53.14m/s,46.5m/s) and plotted a graph of Tension vs. Velocity^2.In investigation 2, we tested the standing waves in an air column which used 3 different tuning forks with 256Hz, 480Hz, and 512Hz to test the first two resonance of each fork. After this, we length of the first and third resonance to measure the wavelength which are 0.6m, 0.664m, 1.332m.and plotted a graph of wavelength vs. 1/f. Introduction There are many different types of waves, such as sound waves, light waves, and other forms. A wave is made up of successive peaks and troughs, and travels in a certain direction. The motion of the peaks and troughs is divided into two types "transverse" and "longitudinal". "Transverse" means that the motion of the crests and troughs is perpendicular to the direction of wave propagation. "Longitudinal" means that the direction is horizontal. According to the relationship between wavelength, velocity, period and frequency, we can get the following expressions: λ V = = fλ T

The vibrations of the strings produce sound and this sound changes by the different tensions of the strings. The speed of the wave along the string depends on the string tension Fs and the mass per unit length of the string μ. The relationship is 푠 푠푡푟 = The speed of sound in air depends on the atmospheric pressure p, the air density ρ and a constant γ. Since we could not calculate it in this experiment, we put the speed of sound at about 343 m/s. After some background on waves, we will move on to the purpose of our experiment - standing waves. Suppose a train of waves traveling through a medium reaches a boundary beyond which it cannot propagate. At such a boundary, the wave will be reflected back in the direction from which it came. The reflected wave will be superimposed on the incident wave. The result is the formation of a standing wave. When two sine waves of equal amplitude and wavelength move towards each other, their superposition is a stationary sine wave with nodes and antinodes as shown in the F0.1 below. This wave is called a "standing wave". F0.1Displacement of a standing wave for five instances in time F0.2 The three lowest string modes After we understand what a standing wave is, we also need to know how to calculate the wavelength of the standing wave. The key to the formation of a standing wave is that there must be nodes at the two fixed ends of the string. However, as shown in Figure F0.2, they can also have additional nodes at the midpoint. The graph at the top of F0.2 shows the longest wavelength. If there are no additional nodes between the two endpoints of the graph, and the length of the string is L, then the longest wavelength of this standing wave is λ1 = 2L. Thus, we obtain the equation of the wavelength on this string is λ n = 2 n L . Once the wavelength is known, the velocity can be calculated and compared to the predicted value by using Vstring formula. Investigation 1 In investigation 1, the setup that we used are 120 Hz vibrator, 2 rods, 3 clamps, 1 pulley, short rod with string clamp, slotted weights, plastic and metal washers, string, paper clip, plastic pail, meter stick , and digital scale.

L did not have a error. The error of the velocity should be V . The equation of the error of V^2 is given in the manual. After finished the calculation, we did the same steps in 4,5,6,7, and 8 nodes. The data we measured is in the table below. T1.1 Properties of wave in string with error for different nodes Nodes 3 4 5 6 7 8 Weight(kg) 0.2968 0.1678 0.1034 0.0738 0.0603 0.0501 Length(m) 1.55 1.55 1.55 1.55 1.55 1.55 error of Length(m) 0.005 0.005 0.005 0.005 0.005 0.005 Distance between two nodes(m) 0.516 0.3875 0.31 0.35 0.2214 0.19375 Wavelength(m) 1.0333 0.775 0.62 0.5167 0.4429 0.3875 error of wavelength(m) 0.003333 0.0025 0.002 0.001667 0.001429 0.00125 Force Tension(N) 2.9086 1.6444 1.0133 0.7232 0.5909 0.4910 error of Force Tension(N) 0.0049 0.0049 0.0049 0.0049 0.0049 0.0049 velocity of string (m/s) 124 93 74.4 62 53.143 46.5 error of velocity(m/s) 0.3871 0.3871 0.3871 0.3871 0.3871 0.3871 velocity squared (m/s^2) 15376 8649 5535.36 3844 2824.163 2162.25 error of velocity squared (m/s^2) 96 72 57.6 48 41.143 36 Then, we plotted the tension F in the string vs. V^2 of the string. F1.2 Tension vs. Velocity^2

F rom F1.2, combined with the previous equation ( 푠푡푟 the mass per unit length of the rope which is 0.0002 kg/m. The value that was given is 0.32g/m. The percentage of difference between the measured value and the given value is − 퐸푥푝푒푟 竡푒 푡 퐺 푒 Percentage of difference = 푥 100%= − 37.5 퐺 푒 This is undoubtedly a huge difference. The main reason that we think the error happened is the difference between the assume value of the weight. For the given value of the weight, we have 900g for 3 nodes but the experiment value that we used for 3 nodes is 296.8g. The huge gap between the value of weight should be the reason why we did not get the similar value of µ . Investigation 2 In investigation 2, we used a completely difference equipment. At this time, we only used sound wave apparatus, and 3 tuning forks with different frequencies. In this investigation, we first add the sound wave apparatus to the right amount of water (about 90%). We then controlled the height of the water surface by means of a reservoir connected with a flexible hose. This changes the length of the air column on the water surface, L. The forks are used to test and thus find the wavelength. The concept is as shown in F2.1, where the first three standing pressure waves in the air column are shown. The figure also shows the relationship between the length of the air column and the wavelengths of these three waves. F2.1 Wave structure of different length = 푠 ), weobtain th e measured valueof µ ,

480 1 0.2 0.005 0.6 0.01616 1/f 3 0.5 0.005 0.00208 5 0.875 0.005 Frequency(Hz ) reasonance average distance between reasonance (m) error of average distance(m) wavelength(m ) error of wavelength(m ) 512 1 0.166 0.005 0.664 0.02108 1/f 3 0.498 0.005 0.00195 5 0.83 0.005 Frequency(Hz ) reasonance average distance between reasonance (m) error of average distance(m) wavelength(m ) error of wavelength(m ) 256 1 0.33 0.005 1.332 0.02126 1/f 3 0.996 0.005 0.0039 5 N/A N/A Based on this data, we plotted a graph of wavelength and frequency. F2.2 Standing wave in air

The slope we got from F2.2 is 368.28. By using math, the slope of the graph is equal to = λf = V 1/f which means the velocity of the air that we got is 368.28m/s. By using the IPL Straight Line Calculator, we obtained the uncertainty of the slope is 13.263. However, the given speed of sound in air is 343m/s which is not included by the uncertainty. The difference is small. The reason that cause it may be the unreasonable wavelength of the 512Hz fork since the larger the frequency is the smaller wavelength it should have. However, the 512Hz fork ’ s wavelength is smaller than the 480Hz fork ’ s wavelength. This small difference may cause the error of the result. Conclusion In conclusion, in this experiment, we learned the propagation of standing wave in different media by two different experimental methods and obtained a lot of standing wave related data by observation and calculation. Finally, we compare the data to determine the success of our experiment. In investigation1, we first tested the conduction of the standing wave on the string. An electromagnetic vibrator and a plastic barrel were attached to each side of the string. The wavelengths of the different nodes were created by changing the weight in the plastic bucket. After testing and calculating the wavelengths, we calculated the velocity and mass per unit length of the rope from the known equations and compared our calculated mass per unit length of the rope with the experimentally given data. Although the difference in the final data is a bit large, we believe that the main reason is that we actually need to generate different weights for the nodes and the experiments give us a large difference. This is the reason for the error. In Investigation 2, we used a very different apparatus than in Investigation 1 to conduct the test. We tested the position of standing wave generation by changing the level of water in the device. We reduced the error in the location of the loudest sound by taking the average value. Then we reduced the wavelength error by the difference between the two positions. Finally, we made a graph of the calculated wavelength and frequency. The slope of λ vs. 1/f is used to calculate the speed of sound propagation in the air from our test and compare it with the experimentally given value. Although there is a difference in the results, the difference is not very large. We think that the main reason for this difference is that the wavelength of our 512Hz fork does not match the principle, so the error in the result is caused. I think in the next experiment, we can be more demanding and precise about the data recording and testing methods. Because our process is correct but the results do not conform to the corresponding logic, there will always be some deviations. λ