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HERE
Laboratory 11
Sou
n
d
Introduction
One aspect of superpositioning is that the amplitude of the resulting wave can be many times greater than that of the original waves. This can be a positive aspect, such as when an ampitheater is built to magnify the sound of performers on a stage without using electronic assistance (which can distort the quality of the sound). It can also have a negative aspect, such as when rouge waves are generated in the ocean that disrupt shipping.
Resonance occurs when the waves produced within a (semi-) closed region are reflected upon each other so that standing waves are produced. At the areas where the underlying waves constructively interfere with each other, the amplitude, and thus the intensity, or the standing wave is increased. In this lab, we will use the phenomenon of resonance to determine the speed of sound in the laboratory.
Additional background:
Imagine two waves of identical wavelength and amplitude traveling in opposite directions with equal speeds. The net displacement of the medium at any point and at any time is determined by applying the superposition principle, which states that the net displacement is given by the algebraic sum of the two individual displacements. The resulting wave pattern will then have points, separated by one-half wavelength, where the displacement is always zero. These points are called nodes. Midway between this nodes, the particles of the medium located at the antinodes, vibrate with maximum displacement.
We can visualize transverse standing waves on a string, of length L, fixed at both ends. These waves can be established by plucking the string at some point and are caused by continual reflection of the traveling waves at the boundaries, in this case the two fixed ends. The boundary conditions demand that at each end there must be a node. We can therefore fit an integral number of half-wavelengths into the length L of the string as shown in Fig. 1.
Even though the standing wave does not appear to be moving, we know that the waves underlying it move with a velocity, v, determined by the properties of the medium. The wavelengths of the traveling waves that combine to give the standing waves must fit within the overall length L such that the fixed ends correspond to nodes in the standing wave. Looking at Figure 1, we immediately see that
L=
nλ
n
2
where n is an integer that has values n = 1, 2, 3, …. Solving this for the wavelength shows that
λ
n
=
2L
n
,
in other words, only certain wavelengths will result in a standing wave. Using the relationship v = f, where v is the speed of the underlying wave, the allowed frequencies are thus
f
n
=
v
λ
n
=
vn
2L
.
Defining the fundamental frequency as
f
0
=
v
2L
,
we can write
fn = nf0.
The frequencies are known as the resonance frequencies of the system. As can be seen from above, the lowest frequency of the system, having the longest wavelength, is called the fundamental frequency and the mode of vibration the fundamental mode or the first harmonic. The modes of vibration with progressively higher frequencies, are called second harmonic (n = 2), third harmonic (n = 3), etc. Notice that as the frequencies increase, the corresponding wavelength decreases, as expected.
Longitudinal waves can also be set up that create standing waves. The simplist way is to use a tube that is either open at only one end, or open at both ends. When the tube is open at one end, we refer to the tube as a closed air column. When it is open at both ends, the tube is an open air column.
Let’s first consider a closed air column. In this case, the closed end of the pipe has to correspond to a node in the standing wave. The open end of the pipe will correspond to an anti-node. Graphically, this looks like
From the figure we see that the wavelengths have to satisfy
λ
n
=
4L
n
where n can only have odd values, n = 1, 3, 5, …. The corresponding resonant frequencies are then
f
n
=
nv
4L
=nf
0
where
f
0
=
v
4L
.
If we look at an open air column, the situation is slightly different. Now we have anti-nodes at both ends of the pipe:
The resulting allowed wavelengths can be seen to still be
λ
n
=
4L
n
,
with an associated frequency of
f
n
=
nv
4L
=nf
0
where f0 is the same as for a closed pipe. The only difference is that now, both even and odd values of n are allowed, n = 1, 2, 3, ….
The energy of the air at different locations along the sound wave are different. Uisng the following simulation examine what is happening to the molecules of air at the node and antinode:
http://www.walter-fendt.de/ph14e/stlwaves.htm
LAB Directions:
You will be given water, tubes or columns and tuning forks to examine the speed of sound and to investigate the relationships between frequency, speed of sound and wavelength. You will design your own lab with these materials and should consider the following questions to help you focus on a specific question and method of data collection.
Some questions that you should be considering as you develop your lab are:
Guided Inquiry Questions:
1) We know that the speed of sound varies with temperature as
v=v
0
√
T
T
0
, where v0 = 331 m/s, T0 = 273 K, and T is in degrees Kelvin. How can you determine this for the laboratory? Why do I want this value for the speed of sound?
2) There is a frequency marked on the tine of the tuning fork. What does this correspond to? Do you need to assume any uncertainty in this value?
3) How can you measure the wavelengths of the various resonance positions for your closed pipe? What are the uncertainties in these measurements?
4) How can you determine which value for n corresponds to each resonance? Which resonances did you find?
5) What is the average speed of sound associated with each resonance position? What is the uncertainty associated with each of these averages?
6) Can you find a better estimate for the speed of sound from your different resonance averages? If so, how? What is the uncertainty associate with this estimate?
7) How accurate is your best estimate for the speed of sound? What are you using for the basis of comparison?
8) What would you get if you repeated this experiment using an open pipe?
Submit for Credit
· Include the full lab that provides the background/introduction section presenting information that is important in understanding the concepts.
· Be clear what your specific question(s) is(are) for your investigation.
· Provide a drawing, picture, image of your set up and a very clear description of how you collected data, measured data, calculated data.
· Provide a data table with the measurements you recorded. For example:
Frequency (f)
n
L(m)
4L/n
V=(4L/n)f
T= 1/f
· Be sure to include an error analysis and explain the relevance of those errors to your results.
· Provide graphs that show relationships and patterns an ex[plain what the graphs show.
· Include a section that addresses the questions you asked (and most likely others from the guided questions above) and explain the meaning of the results, the lab and how it can be applied in a real world situation.
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PHET Sound & Fourier Simulation
Purpose – To explore sound (a longitudinal wave) and how it can be modeled as a transverse wave
Obtain computer with PHET software on it. Get headphones if possible and plug into sound port.
Open PHET simulations and find the Sound simulation:
http://phet.colorado.edu/en/simulation/sound
Open the simulation called “Sound”. There are 5 tabs.
Open the one called listen to a single source.
1. What is the affect of changing the frequency of the tone generated?
2. What is the affect of changing the amplitude of the tone generated?
3. Using the slide bar as a rough ruler, and the frequency set to 100, how low in amplitude (%) can you still hear the tone?
4. Using the slide bar as a rough ruler, and the frequency set to 1000, how low in amplitude (%) can you still hear the tone?
5. Do you have better “high end”, or “low end” response with your ears?
Design an Experiment :
6. Press “start” and move the ruler to the center of the speaker.
a) Look at the stopwatch. What do you notice that is strange about it? Why is it programmed this way?
b) Describe how you would find the frequency of a wave if the frequency slider did not have a number display. Test your idea with a variety of waves (record them in a data table) and describe how well your procedure gives results that match the frequency display.
c) Describe how you would find the period of a wave without using the frequency information. Test your idea with a variety of waves and record your experiment in a data table. Check your method by calculating the period using the frequency (T = 1/f). Show calculations.
d) Hit stop and reset, and measure the distance a wave travels in a certain amount of time. Make a data table and do at least 3 trials. Find the speed of sound using v = d/t.
e) Use the ruler to measure the wavelength of this sound wave. Check the speed calculated above using v = fλ.
7. Using a experiment of your own design, measure the speed of sound. Record any data you obtain here.
8. Based on the speed you measured above, how long would it take for you to hear thunder if you observed lightning and it was seen to be 1620 meters away?
Open the panel called Two Source Interference
9. What do you suppose the light and dark bands on the “sound” emanating from the speaker represent? Enable Audio, change to Listener mode, not Speaker mode!
10. Move the head of the listener up and down. What do you note about the locations where the waves intersect? Is there a location in which you hear less sound? What do you think is happening? Don’t forget that sound is due to a change in local pressure.
Open the panel called “Interference by Reflection”
11. Explore the affect of changing frequency on the reflected wave angle. Is there any affect?
12. Explore the affect of wall angle and wall position on interference pattern. Can you draw (roughly) a set up that shows interference that would not allow you to hear a certain frequency? In the space below show the speaker, the wall angle, the wall position, and a location where you think the sound would “drop out” due to reflection.
Open the simulation of Listen with varying Air Pressure
13. Leave the pressure set to 1 atm. Adjust the amplitude of the sound (at a fixed frequency), what do you notice? Make sure you’re in listener mode!
14. Adjust the pressure down, and record the relative loudness at 5 data points. Make up your own scale (for loudness), and record the loudness (y-axis) versus pressure (x-axis). Make a rough plot of the data here. What do you think the relationship is?
Go back to the main screen on the Simulations. Choose the Fourier Series simulation:
http://phet.colorado.edu/en/simulation/fourier
Choose the discrete tab
15. Listen to the main tone. What is the affect of adding small amounts of the harmonics (overtones?)
16. For a basic simulation, with up to four harmonics, draw below each wave form and the summary wave form. Use different pens/pencils to represent the different waves (label them).
17. What can you say about the regions where the waves have the same phase (are on the same side of the y-axis)? How do they add up in the summary wave? Show a point on the summary wave and label this “constructive”.
18. What can you say about the regions where the waves have opposite phase (are on opposite sides of the y-axis, or are shifted). How do they add up in the summary wave? Show a point on the summary wave and label this “destructive”.
Open the “Wave Game” – Play the game, and see how high a level you can go before you can’t match the wave.
19. How high did you make it?
20. Comment here on what you learned in trying to match the wave form.
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