Lab 3- Waves

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Feb 20, 2024

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Physical Science 1422 Lab 3: Standing Waves and Velocity of Sound in Air Name: Evelyn Yanez, Joshua Villarreal, Esteffy Velasco, Marissa Zuniga The purpose of this laboratory activity is to study concepts of waves, standing waves and the speed of sound in air. The main concepts in this lab are wave properties of sound, resonance, and the standing wave. Vibrating Strings Introduction: The general appearance of waves can be shown by means of standing waves in a string. This type of wave is very important because most of the vibrations of extended bodies, such as the prongs of a tuning fork or the strings of a piano, are standing waves. The purpose of this experiment is to examine how the tension required to produce a standing wave in a vibrating string of fixed length and mass density is affected by the wavelength and the frequency of the wave. Theory: Standing waves (stationary waves) are produced by the addition of two or more traveling waves, which have the same wavelength and speed, but travel in opposite directions through the same medium. Figure 1 shows such a system, where a mechanical vibrator produces a wave on a string which moves to the right and reflection from a fixed end produces a left moving wave. Where the two waves are always 180 0 out of phase, very little motion occurs (none if the amplitudes are the same). Such places are called nodes (see Figure 1). Where the two waves are in phase, the motion is maximum. These positions are call anti-nodes. Figure 2 shows a representation of a standing wave. This representation shows the two extreme positions of the string. This matches well with what we see since the string speed is minimum at the extremes. However, don’t forget that the string goes through all of the positions in between the extremes as shown by the blur in Figure 1. Note that as shown in Figure 2, the node-to-node distance is one-half of the wavelength. The necessary conditions for the production of standing waves on a stretched string fixed at both ends is the length of the string be equal to an integer number of half wavelengths so that there can be a node at each fixed end. Figure SEQ Figure \* ARABIC 3 - Standing wave representation Anti-Node Node Node Figure SEQ Figure \* ARABIC 1 - Standing wave segment on a string

end and one node in the middle; then the wavelength is equal to the length of the string. It may also vibrate with a larger integer number of segments. In every case, the length of the string equals some integer number of half wavelengths. If you drive a stretched string at an arbitrary frequency, you will probably not see any mode; many modes will be mixed together. But, if the tension and the string's length are correctly adjusted to the frequency of the driving vibrator, one vibrational mode will occur at a much greater amplitude than the other modes. For any wave with wavelength λ and frequency f, the speed, v, is v = λf Eq. (1) The speed of a wave on a string is also related to the tension in the string, F, and the linear density (=mass/length), μ , by v 2 = T/ μ = λ 2 f 2 (Eq. 3) L is the length of the string and n is the number of segments. (Note that n is not the number of nodes). Since a segment is 1/2 wavelength then λ = 2L/n, where n = 1,2,3,… Eq. (3) Solving Equation 2 for the tension yields: T = μ λ 2 f 2 Eq. (4) Linear Density (µ) represents an object’s mass per unit length. We can measure linear density using the following formula: Mass/Length = Linear Density (µ) Eq. (5) In this experiment, we will be finding the tension required to produce a 1 standing wavelength (2 segments) in a string of fixed length and examining the change in required tension as the frequency of the wave is varied. Activity 1: Lab Activity: Standing Wave Vibrating Strings - Lab Report Setup: Use the following website for this part of the lab: https://ophysics.com/w8.html Constant Wavelength () Procedure: In the first part of the experiment, we will hold  constant by always choosing a two-segment pattern so that  = L since n = 2 and vary the frequency while measuring the tension at which a two-segment standing wave appears. By plotting F versus f 2 , we should see a straight line with a slope of μ  2 . 1. Use the following initial settings: ● Linear Density = 0.3 x 10 -3 kg/m ● Frequency = 60 Hz ● Tension = 16 N Figure SEQ Figure \* ARABIC 2 - Modes of vibration

2. You should see 2 oscillating wave segments on your simulation, as shown below. Remember that 2 segments are equivalent to one complete wavelength! The amplitude (height) of the wave is pretty low and will be maximized at the correct Tension value. Figure 4 - Standing waves simulation with initial values 3. Decrease the tension by clicking the slider under “Tension” once and pressing or holding the DOWN arrow on your keyboard. You can also drag the slider slightly to the left. Continue until your frequency is around 15 N. This should decrease the amplitude, which means we are moving away from our desired Tension value to produce a standing wave. Figure 5 - Slider control for simulation. Click and drag to make rough modifications. Hold up/down arrows on keyboard for smaller adjustments 4. Increase your Tension back to its initial value of 16 N and now increase the tension by clicking the slider underneath Tension once and pressing or holding the UP arrow on your keyboard. At the optimal Tension value, the wave amplitude should be much larger , almost reaching the line above. Input this Tension value in Table 1 below . Figure 6 - Appearance of the standing wave at the optimal Tension value 5. While maintaining linear density and wavelength as constants, use the values for frequency in the Table 1 below, and fill the values for frequency squared ( f 2 ) and the String Tension where the amplitude of the wave is largest. Table 1 - Variable Frequency Linear Density, μ (kg/m) Wavelength, λ (m) Frequency, f (Hz) f 2 String Tension, F (N) 0.3 x 10 -3 4 60 3600 17.3 N 0.3 x 10 -3 4 79 6241 29 N 0.3 x 10 -3 4 100 10,000 48 N 0.3 x 10 -3 4 120 14, 400 68. N

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0.3 x 10 -3 4 144.3 20,822.49 100N Constant Wavelength Analysis: Graph the values of F vs. f 2 . The plot should be linear. Find the slope of the line that best fits the data. *Note: You can use PASCO Capstone, Excel or other spreadsheet software that you are comfortable using to complete this task . One easy option is to use the following website: https://www.desmos.com/calculator You can simply follow the steps below to graph and export your data: 1. Highlight the f 2 and String Tension columns of your table. Ctrl+C to copy. 2. Go to https://www.desmos.com/calculator 3. Ctrl+V to paste your data into the box. Your plotted points should automatically appear! 4. Press the “ + “ button in the top left corner, then click “expression”

5. To create a new line of best fit, type in “ y1 ~ mx1+b” in the new field and your line should automatically fit to your data points. Slope is also automatically displayed in this box as “ m ” under Parameters. 6. To export your graph, you can screen shot, or just export graph by clicking the “Share Graph” button in the top-right corner of the graph. Click “Export Image” and download the image to insert into this report. 7. Attach your graph on the last page of this report. Fill in the following values and find the percent difference Theoretical value (from constant values in Table 1): μ λ 2 = ____ (0.3x10^-3)(4^2)=4800 _________4.8x10^-3 Experimental value (from your graphed data points): Linear slope ( m ), of graph of F vs. f 2 = ____ 0.0048 _________ % Error = ______ 0_ _______

Attach your graphs for the lab procedure below. When you have completed the repor t, upload your submission to Blackboard Activity 2: Resonance Tube to determine the Velocity of Sound Use the following link for this part of the lab: http://physics.bu.edu/~duffy/HTML5/speed_of_sound.html to complete this part of the virtual lab. Resonance Tube : The animation for this experiment consists of a long cylindrical plastic tube attached to a water reservoir. The length of the water column may be changed by using the “Alternate Level Adjustment” option in the animation while the tuning fork is held over the open end of the tube. Resonance is indicated by the sudden increase in the intensity of the sound when the column is adjusted to the proper length. In this animation, you will be recognized that condition for maximum sound as soon as you find the suitable length of air column with antinode forming at the open end of the tube. The resonance is a standing wave phenomenon in the air column and occurs when the column length is: λ/4, 3λ/4, 5λ /4 where λ is the sound wavelength.

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The water surface constitutes a node of the standing wave since the air is not free to move longitudinally. The open end provides the conditions for an antinode, but the actual antinode has been found to occur outside the tube at a distance of about 0.6 r from the end, where r is the tube radius. This end correction may be added to get a more accurate value if only one resonance can be measured, but it is usually more convenient to eliminate this "end effect" by subtracting the resonance length for λ/4 from those for 3 λ /4, 5λ /4, etc. For this virtual lab purpose, we will not apply the end correction. Procedure : 1. Choose the tuning forks (440 Hz) and produce the 1 st and 2 nd resonance condition as shown in below in the figure and the table 1. Table 1 has been filled for you for the tuning fork with frequency 440 Hz. First reproduce the similar condition and understand what is happening when resonance condition is met.

2. Using the water level, surface slowly to obtain the 2 nd and 3 rd resonance condition if length available. As soon as antinodes form as shown for second resonance, stop there and record the length in m in the table below. When a resonance is found, antinode will form at the open end of the tube. Move the water surface up and down several times to locate the antinode exactly at the open end of the tube. 3. Lower the water further to find the next resonant length. Continue in this manner as far as the length of the tube will permit. Obtain the lengths λ/4, 3λ/4, etc. in meters from your measurements. You will need to check to see if your column lengths follow the progression 1, 3, 5, 7, -- since you may have missed a resonance or counted one of the fainter spurious resonances which sometimes occur. Calculate the wavelength and velocity of sound. 4. Repeat the procedure for the other tuning forks (512 and 660 Hz). The velocity in miles per hour may be found by multiplying the velocity in m/sec by the factor 2.24. Please assume the that you are doing an experiment at 0 °C for reference since the velocity of sound increases with increasing air temperature. Simulation Hint : This is a simulation of a standard physics demonstration to measure the speed of sound in air. A vibrating tuning fork is held above a tube - the tube has some water in it, and the level of the water in the tube can be adjusted. This gives a column of air in the tube, between the top of the water and the top of the tube. By setting the water level appropriately, the height of the air column can be such that it gives a resonance condition for the sound wave produced by the tuning fork. In the real experiment, resonance is found by listening - the sound from the tube is loudest at resonance. In the simulation, resonance is shown by the amplitude of the wave in the air column. The larger the amplitude, the closer to resonance. Note that at certain special heights of the air column, no sound is heard - this is because of completely destructive interference. In addition, there is always a node (for displacement of the air molecules) at the water surface. To a first approximation, resonance occurs when there is an anti-node at the top of the tube. Knowing the frequency of the tuning fork, the height of the air column, and the appropriate equation for standing waves in a tube like this, the speed of sound in air can be determined experimentally. What do you get for the speed of sound in air in this simulation? Use the following link for this part of the lab: http://physics.bu.edu/~duffy/HTML5/speed_of_sound.html Data Table

Frequency Of the tuning fork Record 1st Resona nce Point (1/4) λ (m) Record 2nd Resonance point (3/4) λ (m) 3 rd resonance (5/4)λ (m) Calculat e Wav eleng th (λ (m)) Velocity of sound v= f λ (m/sec) Velocity of sound v= f λ (miles/hr) 440 0.19 m 0.57 m ------------- 0.76 334.4 750 512 0.17m 0.50m ------------ 0.67 343.04 767.4 660 0.13m 0.39m 0.64m 1.16 343.2 789.4 Questions 1. Calculate the Average Velocity of Sound in (m/s) 340.2 is the average velocity of sound. 2. Velocity of Sound in Air Sound travels at 331.5 m/sec in air at 0°C. As the temperature of the air increases, the speed of sound also increases. The velocity of sound at any temperature T, can be calculated using the following linear equation. V T = (0.6 * T) + V ex (0°C) (a) Calculate the velocity of sound at 30 0°C and 100 0°C, where V ex is the average velocity of sound you calculated during your virtual experiment or Q1 or (table above) V T = (0.6 * 30°C) + 344 (0°C)= 358.2 m/s V T = (0.6 * 100°C) + 344 (0°C)= 400.2 m/s 3. Using the Velocity of Sound at 331.5 m/sec in air at 0°, as your accepted value and the average velocity of sound as the experimental value. What is the percent error? Percent Error= 2.624% Reference: 1. http://physics.bu.edu/~duffy/HTML5/speed_of_sound.html and http://hyperphysics.phy- astr.gsu.edu/hbase/Class/PhSciLab/restube2.html

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Lab 3- Waves

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