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Standing Waves
Equipment
Qty Items Parts Number 1 String Vibrator WA- 9857 1 Mass and Hanger Set ME- 8967 1 Pulley ME-9448B 1 Universal Table Clamp ME-9376B 1 Small Rod ME- 8988 2 Patch Cords 1 String
Purpose
The purpose of this activity is to examine the relationships between the tension in a sting, the length density of the string, the length of the string, and the standing waves that can form on the string.
Theory
A standing wave is a wave where the overall pattern does not appear to move. For standing wave to form on a string, the basic condition that must be met is that both ends of the string must be fixed in place, never moving themselves. This means that for a string stretched out horizontally, the vertical displacement of each of the two ends of the string must always be zero! In a standing wave, the locations of the string that never move are called nodes, while the locations on the string where the vertical displacement will reach its greatest value are called antinodes. Using the condition that there must be a node located at each end of the string for a standing wave to form on it, let us construct a relationship between the length of the string and the size of the waves that can form standing waves on that string. Let the length of the string be ๐ฟ, then the largest wavelength that will allow for our condition to be met is a wave that is twice as long as the string itself, ๐ = 2 ๐ฟ which will result in a standing wave with 1 loop to form on the sting. rev 05 /201 9
The speed of a wave ๐ฃ is given by the product of its wavelength ๐, and its frequency ๐. ๐ฃ = ๐๐ Inserting 2 ๐ฟ in for the wavelength, and then solving for the frequency gives. ๐ 1 =
This equation gives us the fundamental frequency, or the first frequency that will cause a standing wave to form on the string. This frequency is also called the first harmonic. Now let us repeat this process to find the second largest wavelength that will form a standing wave on the string of length ๐ฟ. The second largest wavelength that will meet the condition of nodes being at both ends of the string is one where its wavelength is equal to the length of the string itself ๐ = ๐ฟ, which will cause a standing wave with 2 loops to form on the string. Inserting this into the equation for the speed of a wave, and solving it for the frequency we obtain: ๐ 2 =
which is the second frequency that will form a standing wave on this string, aka the second harmonic, aka the first overtone. Repeating this process one more time to find the third largest wave that will form a standing wave on the string of length ๐ฟ. The wavelength of this wave will be related to the length of the string by ๐ = 2 ๐ฟ 3 , which will cause a standing wave with 3 loops to form on the string. Inserting this into the equation for the speed of a wave, and solving for the frequency gives, ๐ 3 =
๏ท Click on the push pin icon near the top right of the signal generator window to rescale the main window so you can keep the signal generator open while you perform the experiment.
- Use two patch cords to connect Output Source Ch (1) to the string vibrator.
- Using the provided equipment construct the setup as seen in the provided picture. ๏ท The length between the string vibrator, and the detachable pulley should be AT LEAST one meter. ๏ท The length of the string should be horizontal. ๏ท Donโt plug in the string vibrator yet. ๏ท The C-clamp needs to be tight enough that it will hold the string vibrator in place, but not too tight that you crack the plastic case of the string vibrator. ๏ท At this point the exact mass hanging from the hook is unimportant.
- Using a measuring stick, measure the length ๐ฟ of your string. Please note that ๐ฟ is not the entire length of the string, but the distance between the pulley and the front edge of the metal โbladeโ the string is tied to. Record your value for ๐ฟ in the table for string.
- Take a long length of the same type of string you are using, 3 to 4 meters, and using a mass scale measure the mass ๐ of the string. Then record that value in the table provided. ๏ท Using a measuring stick measure the length ๐ of this same string. Then record this value in the table for string. ๏ท Using your values for ๐ and ๐, calculate the length density ๐ = ๐ ๐ of the type of string you are using, and then record that value in the table for sting.
Procedure: Part 1
- Including the mass of the mass hanger, have 100 grams hanging from the end of the string.
- In the signal generator window set the frequency of ๐ = 60. 0 ๐ป๐ง, then set the amplitude to 1 V.
- In the signal generator window, click the on tab to start the string vibrator to start vibrating.
- Change the frequency till you find the frequency that allows for one standing loop, n = 1, to form on your string. Record this frequency in the table provided. ๏ท You will have to turn the string vibrator off, and then back on to reset the frequency each time.
- Find the frequencies that yield the standing wave patterns that correspond to n =2, n = 3, n = 4, n =5, and record them in the table provided.
๏ท In the signal generator window, you will have to increase the amplitude of the wave to better see the standing wave patterns for the higher frequencies.
Procedure: Part 2
- In the signal generator window, set the frequency of ๐ = 60. 0 ๐ป๐ง.
- In the signal generator window click the on tab to start the string vibrator to start vibrating. ๏ท Adjust the mass on the hook by adding or subtracting mass till a standing wave of 2 loops form on the string. (The exact mass will depend on the length ๐ฟ and the length density ๐ of the string you used.) ๏ท Once a standing wave of 2 loops has formed on the string, record the total hanging mass as ๐ 2 in the chart provided. Remember the hook itself is 5 grams and needs to be included in the total mass.
- Calculate the tension, ๐, the force in the string created by the hanging mass, then record that tension in the table provided.
- Calculate the speed of the standing wave using the equation ๐ฃ = โ ๐ ๐, then record that as the theoretical speed ๐ฃ๐ in the tables provided.
- Calculate the experimental wave speed by using the equation ๐ฃ = ๐๐, then record the experimental speed ๐ฃ๐ธ in the tables provided. ๏ท The wavelength ฮป is always the total length of 2 loops.
- Calculate the % error between the theoretical and experimental values of the speed of the standing wave with 2 loops.
- Repeat Step 4 through 8 for 3, 4, and 5 loops forming on your string, and record the total hanging mass for each case in the provided tables.
Analysis of Standing Waves Lab
Name______________________________________________ Group#________
Course/Section_______________________________________
Instructor____________________________________________
Table for String (5 points)
Table Part 1 ๐ = ๐๐๐ ๐ (5 points)
Table Part 2
๐ = _๐๐ ๐ฏ๐________
%error
Sample Calculations for table 2: (15 points)
1. For part 1, calculate the theoretical frequency for the standing wave that corresponds to
n = 1 (5 points)
2. Calculate the % error between your experimental frequency for n = 1, and the theoretical
frequency. (5 points)
3. According to the theory all the higher frequencies that form standing waves on a string,
given identical conditions, should all be whole number multiples. Does your data support
this theory? If not, what are some reason why it doesnโt? (5 points)
4. From the data from table 2, and using Excel or similar program, graph T (tension) vs n,
with the trendline, and show equation on the graph. Describe the shape of the graph.
(10 points)
5. Using algebra show that the tension can be written as ๐ = ( 4 ๐๐^2 ๐ฟ^2 )^
1 ๐^2
(5 points)
