Standing Waves in a String

Standing Waves in a String OBJECTIVE To understand the circumstances necessary to produce a standing wave. To observe and define the quantities associated with a standing wave. To determine the wavelength of a given standing wave and relate that wavelength to the string's tension. To experimentally determine the frequency used to produce a standing wave. INTRODUCTION When a string, fixed at both ends, is plucked, waves travel down the string and are reflected at both ends. Under the right conditions, these reflections will interfere with each other to form standing waves. These standing waves are characterized by alternating patterns of nodes (stationary places, along the standing wave, where destructive interference took place) and antinodes (places, along the standing wave, that have maximum amplitude due to constructive interference). The standing waves are defined by the length, tension, and mass density of the vibrating string. Experimentally, a string vibrator will provide the constant frequency and hanging masses, under the influence of the acceleration due to gravity, will provide the tension. APPARATUS (1) electric string vibrator, (1) 0-6 VDC Power Supply, (1) eight-foot length of string, (1) digital balance {for class}, (1) meter stick, (1) mass hanger, (1) slotted mass set, (1) pulley with table clamp, (2) wires and (1) C-clamp. THEORY Standing waves are produced when a resonance exists between the natural frequency of a string and the frequency of the disturbance. When two waves of the same frequency and amplitude traveling in opposite directions meet, they superimpose upon each other. This superposition results in constructive and destructive interference between the two waves. The alternating pattern of constructive and destructive interference defines the standing wave. The periodic destructive pieces in the standing wave are called nodes, while the periodic constructive interference points are called antinodes. A wave is one means by which energy is transported. A wave, in general, is characterized by its wavelength λ [m], its frequency f [Hz], and the velocity of the wave v [m/s]. These variables are related by the expression: v = f Equation 1 Standing Waves in a String - Page 1

A standing wave in a string is called a transverse wave, one in which the displacement of the wave is perpendicular to the velocity of the wave. The magnitude of the displacement, from equilibrium, is called the wave's amplitude A. Typically, the larger the displacement, the more energy the wave carries. The inverse of the frequency of the vibration (number of vibrations (cycles) per second) is the period T [s] (time to complete one vibration (cycle)). A typical transverse wave is illustrated in Figure 1 below. Displacement λ λ + A Figure 1 - A Position or Time T Where, λ [m] is the wavelength (noting that it can be measured between two different points). The wavelength is the physical linear length of one vibration. A [m] is the amplitude of the wave (the positive and negative signs indicated the relative position of the wave from equilibrium) and T [s] is the period of one vibration. Figure 1 can be represented as a standing wave by fixing the vibration of the wave at both ends. Figure 2 illustrates this. λ/2 /2 Antinode Node Equilibrium Figure 2 Fixed End Fixed End Length of the String Note the appearance of the standing wave is to give the illusion that the wavelength measure is the distance from the first peak (antinode) to the second peak (antinode), as was illustrated in Figure 1 of the transverse wave. Due to the constructive and destructive interference patterns, the distance between every other peak (antinode) is the true wavelength of the standing wave. The solid line, of Figure 2, is the initial wave disturbance and the dashed line represents the reflection at either of the fixed ends. Standing Waves in a String - Page 2

A string, fixed at both ends, must have a node at both ends. Further, there exists an antinode between every pair of nodes. Thus, only an integer number of halfwavelengths can fit between the ends. Based on Figure 2, the experimentally measured distance between two nodes is one-half the wavelength of the standing wave; where the pulley is considered to be a node as well (we would call this the experimental wavelength). (Node to Node Distance) 2 = λ Equation 2 Theoretically, in terms of the length of the string L [m], defines that a standing wave will occur for: L= n 2 ; n=1,2,3,4,... Equation 3 Since, λ [m] can be calculated based on the standing wave number we call this λ [m] the theoretical wavelength. Figure 2 represents the n = 3 standing wave (L = 3λ/2). The n = 1 and n = 2 standing waves are pictured below. Figure 3 Standing Waves in a String - Page 3

As the frequency of the standing wave is constant and using Equation 2 to show that the wavelength of the standing wave changes, Equation 1 indicates that the velocity of the standing wave also changes. The speed of the wave is also affected by the tension in the string and the mass density of the string, given by the relation: v = F ) = F μ L o ( m o Equation 4 Where, v [m/s] is the wave's velocity, F [N] is the tension in the string, mo [kg] is the mass of the string, and Lo [m] is the length of string used to determine the mass mo. The quantity mo/lo is called the linear mass density, μ, of the string. Changing the tension changes the velocity of the wave. As frequency is a constant, a change in the tension will result in a different number of standing waves. This relation can be stated by the following equation (derived from Equations 1, 3 and 4): f ( v = = ( F ) 2 L n = ) n 2 L F ; n= 1,2,3,... Equation 5 In this experiment, the frequency is fixed. Equation 5 can be used to solve for the tension needed to observe a certain standing wave number. But, it is noted from Equations 1 and 4 we get: = v f = 1 f F Equation 6 Where, the wavelength of the standing wave can be varied by directly changing the tension in the string. Where, the frequency and mass density are constants in the experiment. Standing Waves in a String - Page 4

The frequency is provided by an electric string vibrator, and the tension is provided by the acceleration due to gravity acting on a set of hanging masses, illustrated in Figure 4. Pulley Vibrator Table String Figure 4 Hanging Masses mg = Tension EXPERIMENTAL PROCEDURE a) Obtain a section of string (as used in the experiment) and determine its mass and length. b) Construct the experimental arrangement as illustrated in Figure 4. The distance from the clamped pulley to the clamped vibrator should be approximately one to two meters. The vibrator clamp should be placed near the front edge of the base not near its rear. Be sure there is enough string to pass over the pulley allowing the mass hanger to clear the pulley. Do not set the hanger too low or it will hit the laboratory table. Note that the pulley should lie along the same line as the axis of the vibrator. If not already present, place a loop in the string's end to which the mass hanger will be attached. c) Measure the length of the string between the vibrator's end and the top of the pulley. Record this value as L, the length of the string. In the Questions section of the laboratory report, indicate WHY this particular length should be used for L and not the "entire" length of the string (from the vibrator to the hanging masses). d) Attach the vibrator to the power supply, turn on the power supply, and set the voltage to approximately 5.0 volts. Add enough mass (to the nearest 10 g) to the hanger until an n = 4 standing-wave is produced. Be sure the mass listed gives the maximum amplitude of the antinodes in the standing waves. Either gently pulling down or lifting up on the string will give an indication of whether mass should be added or removed from the hanger. Standing Waves in a String - Page 5

e) The distance from the top of the pulley to the first node is significant to your investigations and you may find it necessary to keep track of this value during each step. In the Questions section of the laboratory report, indicate why you suppose this is true? f) Now adjust the system to produce an n = 5 standing-wave. g) Repeat your procedure and data collection for the standing-wave-cases n = 6 through n = 10; if possible. COVER PAGE REPORT ITEMS (To be submitted and stapled in the order indicated below) (-5 points if this is not done properly) Completed Laboratory Responsibility and Cover Sheet DATA (worth up to 20 points) Data tables are available as a downloadable Excel file DATA ANALYSIS (worth up to 20 points) The SIX required sample calculations, to be shown on a separate sheet of paper in your laboratory report (NOT on the data table sheets themselves), are highlighted in yellow on the downloadable Excel data table spreadsheet Standing Waves in a String - Page 6

GRAPHS (worth up to 10 points) With the known values of the experimental wavelength and tension, Equation #6 can be written in y = mx + b form (shown below): =( f 1 ) Graph Equation F +0 Therefore, open a blank Logger Pro file and make a graph with the experimental wavelength on the y-axis and the square root of the tension on the x-axis based on the Graph Equation above. Be sure to place a best-fit-straight-line on your graph using the regression button. GRAPH ANALYSIS (worth up to 10 points) Based on the basic form a linear equation (y = mx + b), which quantities in the Graph Equation above correspond to: The slope? The y-intercept? Using only the variables indicated in the Graph Equation, and showing a complete set of steps, solve the SLOPE part of Graph Equation above for the frequency f. Your final equation must be in equation for the frequency; which must include the variables indicated for the slope part of the Graph Equation as well as the value of the slope from the graph. SLOPE # = (solve for f) ** Solve this for f, the "graph experimental frequency." ** CONCLUSION (worth up to 20 points) See the Physics Laboratory Report Expectations document for detailed information related to each of the four questions indicated below. 1. What was the lab designed to show? 2. What were your results? 3. How do the results support (or not support) what the lab was supposed to show? o For example, you might want to say something like: "From equation # I'd expect that the will because. Based on the data I collected I found that ; thereby." 4. What are some reasons that the results were not perfect? Standing Waves in a String - Page 7

QUESTIONS (worth up to 10 points) DO NOT forget to include the answers to the TWO questions that were asked within the experimental procedure 1) Equation #5 can be written in the form: f = n 2L mg μ Here we have replaced the tension variable with the factor that actually caused the tension..."mg." Showing all steps, solve this equation for the mass, m. Have done this, and using the "graph experimental frequency," calculate the mass necessary to produce the n = 1, 2, & 3 standing-wave-cases. Why do you suppose that these cases were not studied? 2) How would the results of the experiment differ if a lighter string were used? BE VERY SPECIFIC!! What about a more massive string? Standing Waves in a String - Page 8