Standing Waves
Introduction Wave patterns can be found in many facets of everydaylife: sound emitted from your car stereo or iPod, electromagnetic radiation emitted by the Sun (you know it as sunlight), even the manifestations of waves traveling beneath the Earth’s surface can make their presence known…as an earthquake.
Equipment • Oscilloscope • PASCO Mechanical Driver • Beaded-chain and beaded-chain/string lines w/attached weight holders • Various flex weights • Function Generator • Various Leads • Metal Rod • Power Supply
Background Standing waves are a resonance phenomenon created whenever waves are reflected back along their incident path so that the incident and reflected waves have a specific phase relationship. Boundary conditions in the medium dictate the frequencies at which standing waves can be established. They exhibit places of zero amplitude, called nodes, where the incident and reflected waves always have opposite phase and hence interfere destructively. The separation of nodes is one-half of a wavelength. Midway between adjacent nodes are the anti-nodes, places of maximum amplitude, where the incident and reflected waves interfere constructively.
Consider a chain or string of length L. If the chain is clamped or tied so that neither end can move freely, the boundary conditions require that the displacement nodes exist at both ends of the chain. It will therefore exhibit standing waves with 0,1,2… nodes in addition to the ones at the ends. These situations correspond to wavelengths of 2L, 2L/2, 2L/3,…2L/(n+1) where n is the number of nodes on the string not including the ones at the ends.
When a chain has one end fixed and the other is free to move, it has a displacement node at the fixed end and a displacement anti-node at the other. In this case, we can have standing waves when λ = 4L, 4L/3, 4L/5,…4L/(2n+1). We can approximate this situation by applying tension to the chain on one end and attaching a much lighter string to the other to simulate the “free” end of the chain.
The propagation speed (v) of a transverse traveling wave on a chain under tension depends on the tension in the chain (T) and the linear mass density of the chain (µ):
T v = µ
If the wave is sinusoidal, the propagation speed is related to the frequency, f, and wavelength, λ, by
! v = f"
The lowest frequency at which a standing wave can be established is called the fundamental frequency. As you will observe, standing waves can also be established at higher frequencies that correspond to specific multiples of the fundamental frequency. These higher frequencies! are called harmonics.
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Procedure The following experiments will allow you to investigate standing waves on strings and the propagation speed of waves on strings. Each table has a setup for both the standing wave and the propagation speed components of this experiment. Be sure to complete both parts of the study; however, they can be completed in any order.
Standing Waves on a Chain Fixed at Both Ends You are provided with both a uniform chain and a chain/string combination. The linear mass density of the chain is µ = 0.0162 kg/m. They are set in motion by a mechanical driver (by PASCO) near one end. The driver is basically the motive part of a loudspeaker, and it is powered by an oscillator-amplifier that generates and amplifies several waveforms (of which we use the sine). The point at which the driver is attached is not exactly a node because it undergoes small amplitude oscillations. The true node lies a small distance away and may be impossible to observe directly, but none of our measurements will be made to that point. The goal of this study is to determine and observe the frequencies at which standing waves are established.
The best procedure is to set the generator to an approximate expected standing wave frequency and then slowly adjust the frequency while observing the chain. When the desired number of nodes can be visualized on the string & the amplitude of the antinodes is maximized, you have located the standing wave frequency. The main problem in locating the desired standing wave patterns arises from the fact that it takes a certain number of cycles for an imposed frequency to die away, and when you turn the dial on the generator to a new frequency, the old one persists for a while and interferes with the new one, producing a formless and confusing appearance. This is why it is important to adjust the frequency slowly.
Predict the theoretical value of the fifth harmonic and then set the frequency of the oscillator to this value, then slowly adjust the frequency to find the actual 5th harmonic.
• In your notebook, sketch figures of the standing wave pattern for the fundamental frequency and the next five harmonics.
• Load the chain line with three kilograms, and estimate the fundamental frequency from f0=u/2L. • Because the higher harmonics are easier to observe, we will begin our observations of standing wave patterns at the fifth harmonic and work our way down to the fundamental frequency. After locating the fifth harmonic record the frequency and continue down the series towards the fundamental, recording your observed stnading wave frequencies along the way.
Note: As you approach the fundamental, it takes longer for non-standing wave frequencies to dampen out. You may find it advantageous after each adjustment to stop the vibrations of the chain altogether with your hand and let them build up again from zero.
Create a plot of the standing wave frequencies vs. the number of nodes (excluding the ones at the ends of the chain), and use a linear fit to determine the slope and intercept. Theoretically, this relationship can be described by f=(n+1)f0, where n is the number of nodes and f0 is the fundamental frequency. Which variables do the slope and intercept of your graph represent?
Standing Waves on a Chain with One Fixed End Repeat the previous study with the half-chain/half-string line. In this case, the boundary conditions have changed, so
you will need to find a different relationship between the fundamental frequency, f0, and the string length. The theoretical prediction for the standing wave frequencies is now f=(2n+1)f0.
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Propagation Speed of Transverse Waves on a String We can measure the propagation speed of transverse waves on the chain with the set-up shown in Figure 1 below by measuring the distance a wave travels along the chain and the time required to travel that distance. In essence, the oscilloscope is acting as a stopwatch, measuring the time interval for a wave to travel along the chain. The wave is created by lightly tapping the chain with a metal rod and detected by the driver unit, which is now working backwards, i.e., it is turning motion into an electrical signal.
The signal is visualized on the oscilloscope, and its position along the horizontal axis is proportional to the time that it took for the pulse to travel from the point where you struck the chain to where the driver is attached.
The metal rod is connected to a voltage source (the power supply) so that the action of tapping the chain, in addition to creating the wave impulse, applies a voltage to the trigger input of the oscilloscope. This action starts a clock on the oscilloscope when the wave is generated.
To recapitulate the sequence of events: 1. The rod hits the chain, simultaneously creating a pulse on the chain and triggering the oscilloscope. 2. The pulse travels along the chain while the trace travels along the horizontal axis of the oscilloscope. 3. The pulse arrives at the driver unit, which transforms it into an electrical signal that appears on the oscilloscope.
Figure 1: Diagram of Wave Propogation Setup
Diagrams have been left on the table that show the settings of the oscilloscope controls. Be sure that your oscilloscope settings agree with those on the diagram before you begin. Tap downward on the chain with the rod. This should cause a trace to appear on the oscilloscope. Try adjusting the trigger level if it doesn’t. The trace should resemble the illustration in Figure 2.
Figure 2: Example of Wave Pattern for Wave Propogation Experiment
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Note that instead of a single pulse, the display appears to be a wave train, making it difficult to decide where to measure timing marks. But it turns out that it doesn’t matter, as long as you consistently use the same reference point to determine propagation time, because, due to the graphing technique wherein the desired parameter is obtained from the slope, the only effect of using a different reference point is to change the intercept.
• Begin by loading the chain with four kilograms (plus the mass hanger, which is 0.15 kg). • Obtain at least five data points by varying the distance the wave travels within a range of 0.1 m to 3.5 m. • Determine the speed of the wave along the chain from a graph of your distance and time data. Compare this result to the theoretical prediction above.
Effect of Tension on Velocity of Propagation • Repeat the above study by loading the chain with masses of 3.5 kilograms through 2 kilograms in half- kilogram steps. • Create a plot of your wave speed and tension data that is linear in the chain tension. • From the graph, extract the experimental value of the chain’s linear mass density and compare it to the value given earlier in the lab.
Concluding Questions When responding to the questions/exercises below, your responses need to be complete and coherent. Full credit will only be awarded for correct answers that are accompanied by an explanation and/or justification. Include enough of the question/exercise in your response that it is clear to your teaching assistant to which problem you are responding.
1. Referring to the sketches of standing wave patterns that you drew for a chain fixed at both ends, determine the number of nodes present (excluding the nodes at the ends of the chain). Give an expression that relates
the length of the chain to the wavelength for each case, and then derive the general expression f=(n+1)f0 that th relates the frequency of the n harmonic to the fundamental frequency, f0. Note that n = 0 corresponds to the case with zero nodes (not counting the end points). 2. Referring to the sketches of standing wave patterns that you drew for a chain fixed at one end and free at the other, determine the number of nodes present (excluding the node at one end of the chain). Write an expression that relates the length of the chain to the wavelength for each case, and then derive the general th expression f=(2n+1)f0 that relates the frequency of the n harmonic to the fundamental frequency, f0. Note that n = 0 corresponds to the case with zero nodes (not counting the node at one end). 3. Consider two chains of equal length and mass density: one is fixed at both ends, while the other is fixed at only one end. How would the tension applied to the two chains have to be adjusted so that both chains vibrated with the same fundamental frequency?
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