Experiment 14 Standing Waves

Report for Experiment #14 Standing Waves Lab Partner: TA: May 17, 2023

Introduction The objectives of the experiment were to study standing waves, in an air column and on a string. The lab’s purpose was to examine the relationship between string tension and wave velocity, as well as to measure sound wave velocity. Waves are a series of successive peaks and troughs travelling in one direction. Standing waves are waves which oscillate through a medium, in this case air and a string, when frequencies interfere with the medium. They are bounded on one side and at this boundary the wave is reflected back to the direction from which it originated. The peaks and troughs of a standing wave often appear to travel over the medium, but the peaks and troughs of the wave merely oscillate. Nodes are the points on the standing waves at which there is zero displacement. The areas of largest displacement are known as antinodes. Waves on a string or in the ocean represent transverse waves, which are formed by up and down motions of the medium. Sound waves that travel through air represent longitudinal waves, which are waves made up of high and low areas of pressure travelling in a direction. Near the peaks the air molecules move towards each other and near the troughs they individual molecules move away from each other. Note that there is no overall motion of the medium, only vibrations back and forth of small components of the waves. Wavelength, shown as λ, which was equal to the distance from one crest to the next or one trough to the next. The time it takes to travel one wavelength or to complete one full oscillation is known as the period, T . The inverse of T is frequency, f , which is the number of oscillations per second. Knowing this and that velocity = Δx/Δt, we can infer that v = λ T = fλ when f = 1 T The velocity at which the wave travelled through air is different from the velocity at which the standing wave travelled through the string, and both can be calculated using v string = √ F s μ v sound ∈ air = √ γρ ρ Where F s is the tension in the string and μ is the mass per unit length of the string. In the second equation, p represents atmospheric pressure, the density of air is shown as ρ, and a constant γ = 7/5. This value was derived from molecular theory. At room temperature and density of air and sea level pressure, the velocity of sound is approximately 343 m/s.

differences between values get smaller. Thus, weight needed to be changed carefully and any mass added or subtracted had to be done so gently. In the table, the distance between each node is recorded and from these values the average distance between two adjacent nodes was calculated and recorded in each trial. The distance between two nodes is equal to ½ λ (wavelength), thus the distance between two nodes, multiplied by two, would equal the wavelength. Tension was calculated by the equation: F s = mg The data table was created with four more columns. One of these included wave velocities ( v string ), which was calculated from the measured wavelength and given vibrator frequency. v = λ T = fλ The error values for v string made up the second column, represented by δ v string . This was calculated using the error of the frequency and wavelengths, via the equation δv v = √ ( δf f ) 2 + ( δλ λ ) 2 The third column contained ( v string ) 2 , calculated by squaring the value of v string . The fourth column contained the error of ( v string ) 2 , calculated by the following equation δ v string 2 = 2 ( v string )( δ v string ) Using the observed and derived data, a graph was made with a plot of tension against v string 2 . The resulting plot, combined with the following equation, provides everything needed to calculate μ, the mass per unit length of the string. v string = √ F s μ Table 1: Observed Data and Calculated Average Distance Between Two Adjacent Nodes

Mass (g) # of Nodes Node 1 Node 2 Node 3 Node 4 Node 5 Node 6 Node 7 Node 8 Average 𝚫 x (m) 960 3 0 0.71 1.46 0.73 492 4 0 0.46 0.96 1.46 0.48 292 5 0 0.365 0.72 1.09 1.46 0.365 192 6 0 0.294 0.59 0.89 1.17 1.46 0.29 132 7 0 0.2 0.42 0.68 0.92 1.19 1.46 0.24 92 8 0 0.19 0.41 0.64 0.84 1.05 1.27 1.46 0.21 The above table displays all five trials and the masses found to cause the desired standing waves. The table also shows the measured distance of each node from the contact point between the string and vibrator. Additionally, the average distance between each node is also shown. It was calculated by measuring the distance between each pair of two adjacent nodes for each standing wave and averaging these values. Table 2: Derived Quantities of Wavelength, Tension, and Wave Velocities Wavelength ( λ ) in meters Error ( δλ ) Tension (N) Error ( δ N) v string (m/s) Error ( δ v string ) v string 2 (m 2 /s 2 ) Error ( δ v string 2 ) 1.46 0.001 9.408 0.0005 175.2 0.12 30695.04 0.0288 0.96 0.001 4.8216 0.0005 115.2 0.12 13271.04 0.0288 0.73 0.001 2.8616 0.0005 87.6 0.12 7673.76 0.0288 0.58 0.001 1.8816 0.0005 70.08 0.12 4911.206 0.0288 0.48 0.001 1.31026 0.0005 50.4 0.12 2540.16 0.0288 0.42 0.001 0.9016 0.0005 49.2 0.12 2420.64 0.0288 The relationship between the distance between two adjacent nodes and wavelength was defined as x = ½ λ . As such, the average distance between two nodes, given in Table 1, multiplied by 2, would equal the wavelength. The error for wavelength was decided to be half of the smallest

Hz. The amount of water in the cup depended on the height at which the cup was held. When the cup was lowered, it filled with water and the water level in the plastic tube lowered. When the cup was raised above the water level of the plastic tube, the amount of water in the cup decreased and more water filled the plastic tube. The water level represents L of the air column directly above the water for the purposes of this experiment. The tuning forks are held over the opening of the tube to excite the air column. Sound waves in the air column have different parameters and conditions than the standing waves on the string. Unlike with the string, the sound waves in the air column will not have nodes at both ends. This is because the top of the air column connected to open air which has constant pressure, producing a pressure node at the opening. At the surface of the water, large pressures can build because the air has nowhere to go. Thus, forming a pressure antinode. Therefore, there is an antinode at one end and a node at the other end of the air column. As illustrated in the figure below. Figure 3: The figure shows the first three standing waves produced by pressure in the air column of the experiment. It also illustrates the relationship between length (L) of the air column and the wavelengths (λ) of the subsequent waves. This relationship can also be represented as λ = 4 L n / 4 n (n = 1, 3, 5, 7, ...). It should be noted that the idea and depiction of a node at the top and antinode at the bottom is quite theoretical and reality is not as cut and dry. The experiment started with the water level in tube high as possible, achieved by holding the reservoir above the top of the tube. A tuning fork (512 Hz ) was held above the tube without touching it and was occasionally struck with a mallet so that it would continue to vibrate. At the same time the reservoir was lowered slowly such that the water level lowered, and L of the air column grew. At a specific point, the air column hit exactly the right length and a sudden increase in sound intensity was observed. This change indicated a resonance point, or a standing wave. If the water level rose a bit or fell a bit more, the intensity and volume of the sound

decreased suddenly. The position of the resonance point ( L 1/4 ) was found a total of three times and recorded, along with the average and the error. The water level was steadily lowered once more in search of the next resonance point (n=3) as shown in Fig. 3. Three separate measurements of the position of L 3/4 were found and recorded in the data table. From the recorded values, the distance between the first and second resonance were calculated. Wavelength was then found by the following equation λ =( L 3 / 4 − L 1 / 4 ) × 2 This was done to take the error out of the calculations for wavelength, because there was systematic error in the uncertainty of the placement and shape or the pressure node at the top of the air column. Then the error for this value was calculated. This entire procedure was repeated for the other two tuning forks with frequencies of 256 and 480 Hz . It should be noted that the third resonance (n=5) was attempted to be found but could not be determined for any of the tuning forks. Using the derived and observed data from the experiment, a plot was made of λ vs. 1/ f with error bars displaying the calculated δλ. A trendline was plotted based on this graph and the slope was derived and its error calculated using the IPL slope calculator. Tuning Fork ( Hz ) Trial 1 Trial 1 λ Trial 2 Trial 2 λ Trial 3 Trial 3 λ 480 0.163 0.46 0.594 0.16 0.468 0.616 0.161 0.463 0.604 256 0.31 0.946 1.272 0.33 0.96 1.26 0.315 0.958 1.286 512 0.134 0.485 0.702 0.129 0.466 0.674 0.132 0.468 0.672 Table 3: The tuning forks and the three trials for each as well as the calculated wavelength. For the purposes of the table, the red heading indicates first resonance, and the blue heading indicates second resonance. All measurements are in centimeters, except for the first column which is in Hz . Tuning Fork ( Hz ) T (secs) Average λ (m) Velocity (m/s) 256 0.003906 1.272667 325.8027 480 0.002083 0.604667 290.24 512 0.001953 0.682667 349.5253

While there was some error in Investigation 1, the values derived from the data align well with given and known information. There is some inconsistency in the data though, which can be attributed to human error. Investigation 2 certainly had more error than Investigation 1. There appears to have been the most difficulty in finding the resonances for the tuning fork of 480 Hz. The calculated velocity from the data is farthest from the other values and the expected velocity. This is likely due to human error and failure to accurately identify resonance points. Questions 1) The lowest frequency of a guitar string with a length 0.65 m is 248 Hz. What is the speed of the wave on this string? The equation for fundamental frequency on a string is f = v/2L, therefore v = 2Lf. v = 2(0.65 m)(248 Hz) = 322.4 m/s 2) Same guitar, same string as in Question 1. If the mass per unit length of the string is 0.5 g/m, what is the tension on the string? 𝑣 𝑠𝑡𝑟𝑖𝑛𝑔 = √ F s μ So v 2 x μ = F s . (322.4m/s) 2 (.0005kg/m) = 51.97088 N 3) A tuning fork produces two maxima, n = 1 and n = 3, separated by 48 cm. Find the frequency of the tuning fork. The two antinodes of the wave are 48 cm apart, so that’s 1⁄2 λ = 48 cm = 0.48m. Speed of sound is 343 m/s. f = v/ λ and 343m/s divided by .48m = 714.583 Hz 4) The speed of sound in helium is 1000 m/s. Suppose you use a tuning fork with a frequency of 512 Hz. What would be the separation of the two maxima (L3/4 − L1/4) in a standing wave apparatus like the one you used in the lab, if the tube were filled with helium gas?

λ = v/f and 1000/512 = 1.953 meters. These two maxima are antinodes so the distance between them is equal to 1⁄2 λ. Therefore, the distance between them is equal to 1.953/2 = 0.9766 meters (0.98m) 5) Using Eq. (14.4), determine how the resonant frequency depends on the air temperature. (Hint: Consider how the density ρ depends on temperature when pressure p is fixed.) v sound ∈ air = √ γp ρ This is the equation the question refers to. Pressure is fixed, therefore ρ is inversely correlated to velocity of sound in air. Air is denser when it is hotter, so temperature and air density are directly correlated such that when temperature goes up, density of air goes up. Therefore, temperature is inversely correlated with velocity squared. v = f λ so when frequency goes up, velocity goes up. Therefore, frequency and air density are inversely correlated. Acknowledgments I would like to thank my wonderful lab partner. He was very helpful throughout both investigations, especially concerning recording the data. I’d also like to thank the TA for being understanding, helping us during our lab and answering all our questions.