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Standing Waves on a String
OVERVIEW:
In this lab you will produce standing waves on a string using a mechanical wave generator. You will study how standing wave patterns fit on a fixed string and how the tension of the string and the frequency of vibration relate in standing wave patterns. PHYSICS PRINCIPLES: Wavelength and frequency Wave speed as a function of tension Standing waves NEW LAB SKILLS: Producing an output signal from a computer interface EQUIPMENT: Mechanical wave driver String Science Workshop 750 interface Power amplifier Weight holder Weights Meter stick Balance (4-arm, to read hundredths of a gram) Computer with spreadsheet and DataStudio software
Introduction
Waves come in many forms, but essentially waves are all vibrations that are able to communicate their energy from one place to another. Water waves, sound waves, and waves on a string don’t actually push water, sound, or string from one place to another, but instead transmit mechanical energy through each respective medium. Several waves can exist simultaneously on the same medium and interact with one another. This allows for interference between waves. This lab will demonstrate a specific interference phenomenon that produces patterns known as standing waves. Any wave can be described by its wavelength, λ, the distance over which a wave completes one cycle. A wave’s frequency, f , is the number of complete oscillations per unit time. For example, if a string goes up and down ten times per second, then the frequency is 10 sec-^1 or 10 hertz (Hz). Generally, the higher the frequency, the shorter the wavelength is. The period ( T ) of a wave is the time for one complete oscillation, or 1/ f. For a traveling wave that travels down a string from one end to the other, we can also talk about the velocity at which the wave travels. Because the wave travels a distance of one wavelength in a time of one period, the velocity, v , is:
v =
distance
time
t
= " f (1)
Imagine that you “pluck” a string so that the disturbance travels from one side of the string to the other as a wave. If you continue to pluck these disturbances, you will set up a series of waves with a frequency that is equal to how often you do the plucking. However, an interesting phenomenon occurs with this string because it is fixed at both ends. This means that the waves will reflect , turn around, and interact with other waves. Waves that must interact in the same place at the same time interfere with one another. Essentially waves will add together so that they exist in the same place so that both contribute to the total amplitude of wave at that specific location. When a string is fixed at both ends it is made to oscillate in many ways, but if it is wiggled at certain frequencies it will "resonate" as a standing-wave pattern (see Figure 1). These patterns occur when the vibrations of a wave and their reflections are perfectly coordinated to produce an interference pattern that stays fixed. Thus, even though waves are traveling back and forth, the patterns of interference between all of these waves create this standing-wave pattern. Each of these potential patterns is called a mode of oscillation. In the simplest, fundamental mode, the whole string moves up and down together. In the next simplest, half of the string is moving up when the other half is moving down. In the next, the string is divided into three segments, alternating up and down, and so on. The number of segments is called the mode number, denoted n. The points between segments that do not move at all are called nodes, while the points that move the most, in the middle of each segment, are designated as antinodes. Standing waves are neat phenomena to look at, but they are also useful because they seem to freeze a wave pattern so that we can easily measure a wavelength. In addition, if you know the frequency that a standing wave is being driven at, you can begin to analyze the nature of different waves and their velocities. For some waves (such as sound waves), the velocity stays constant even when frequency and wavelength are changed; yet for other waves (such as water waves) velocity is dependent upon wavelength. Knowing that a wave’s velocity will be the product of its wavelength and frequency (see Equation 1) can be helpful in determining how a wave’s velocity changes with other variables. Each segment is equal to half a wavelength. With a little bit of thinking and staring at figure 1, you can show that the wavelength, λ, is
2 L
n
where L is the length of the stretched string and n is the mode number, or the number of segments in the string. For your standing waves, the value for L is measured between the mechanical wave generator to the pulley, since this is the distance over which the standing wave patterns are produced.
Making waves
The standing wave apparatus consists of a string with one end attached to a rod or ring stand, and another end from which masses can be hung after being strung over a pulley. A mechanical wave generator is a speaker that has a fitting attached to it so that it can wiggle the Figure 1: The first three modes of a string with fixed ends.
hanger) ranges from 50g to 400g. Masses that are too small will not allow for enough tension in the cord to obtain good results, while masses that are too large will strain the cord too much. To run the mechanical wave generator, you will use the computer and the ScienceWorkshop interface. Although you have used this before, this is the first time that you have used this interface to output data rather than just record data. Do the following to setup the interface to control the mechanical wave driver:
- Connect the ScienceWorkshop interface to the computer, turn on the interface, and turn on the computer (if this has not already been done).
- Connect the Power Amplifier DIN plug into Analog Channel A of the interface. (It is important that you use Channel A, since it is the only output that will “talk” to the Power Amplifier.)
- Connect the banana plugs (the red and black cords) from the power amplifier to the mechanical wave driver. Make sure that the mechanical wave driver is in the “unlock” position.
- Open the file on your computer’s desktop (or in an appropriately labeled folder) named “Waves”. This should open the DataStudio software, along with an appropriate setup to control the power amplifier and mechanical wave driver. To avoid overloading the equipment, do not turn on the power switch of the power amplifier until the rest of the equipment setup is complete.
- With the controls provided in DataStudio (see figure 3), you can now vary both the frequency supplied to the power amplifier (and then to the mechanical wave driver) as well as the “voltage”. Although you have not studied what voltage is yet, you can think of this as a variable of the amplitude of the vibrations. Set the voltage to 4.00 V for all of your experiments for best results. For one of your experiments, you will vary the frequency. A good range for this is from 0 to 250 Hz. Figure 3 : DataStudio controls for the power amplifier and mechanical wave driver. With everything connected and ready to go, you may turn on the power switch to the power amplifier. When you press the “Start” button in DataStudio , it will begin controlling the power amplifier and wave driver. Begin playing with the wave driver, varying the frequency with the controls provided. Before you begin any of your experiments, get a feel for how the equipment functions, and produce a few different standing waves. Make sure that everyone in your lab group understands this process and has a chance to set up a standing wave pattern.
Extracting the Patterns
By now you've probably noticed some patterns in the relation between frequency and wavelength. To bring out these patterns, you could make a plot of frequency vs. mode number, or frequency vs. wavelength. A more physically meaningful graph, though, would be a plot of frequency versus the reciprocal of the wavelength, 1/λ. From equation (1) we have
f =
v
"
so that if you were to plot frequency on the y-axis, and 1/λ on the x-axis, the slope of this graph at any point should represent the wave velocity for the given frequency and wavelength. Furthermore, if the slope of such a graph were the same value for all wavelengths and frequencies, you could conclude that the wave velocity is constant for all values of wavelength and frequency. Using all of this information that has been given to you, you should be able to answer the questions posed in this lab. Use Excel to input and graph all of your data. Work with your lab partners to both design an experimental setup and to collect data. Make sure that you justify all of your answers in the lab report with the appropriate data and analyses. You will be graded for not only your answers, but also your justifications and use of data.
QUESTION 2: For a given tension on the string, what is the wave speed, including your uncertainty? Show your work, as well as the final answer. Include the tension on the cord that you are using. QUESTION 3: If you produce several graphs (or several sets of data on a single graph) of frequency versus 1/wavelength by varying the tension on the string, what does this tell you about the relationship between wave speed and string tension?
QUESTION 4: The linear density of a string is defined as:
mass
length
Using this definition, you can calculate the theoretical wave speed for a wave on a string:
v =
T
where T represents the force of tension in the string. What does a plot of your velocity versus tension data look like? Is it reasonable, given the above relationship? QUESTION 5: If you plot a graph of wave speed versus the square root of tension, what do you expect this graph to look like?
