Light Waves and Colors Legacy Problem #13 Guided Solution
Problem*
Mr. H takes his class to the gymnasium to investigate two point source interference patterns produced by sound waves from two sound sources. The pure-tone output from a frequency generator is split and fed to two audio speakers positioned about 1-meter apart. The sound from the two speakers travels through the gymnasium and interferes constructively and destructively to create a pattern of nodes and antinodes. Mr. H directs the class to stand with one ear facing the speakers and the other ear covered and to walk slowly across the gymnasium, observing positions of relatively soft and loud sounds in alternating fashion. Once initial observations are made, Mr. H asks all the male students to stand at nodal positions and all the female students to stand at antinodal positions. Once done, Mr. H takes a picture of their positions and then makes several measurements of the distances between students and from some selected students to the speakers. Some sample data are shown below. Complete the table, determining the wavelength of the sound waves based on each student’s measurements.
| Student | Nodal or Antinodal Line | Distance to Speaker 1 (m) | Distance to Speaker 2 (m) | Wavelength (m) |
|---|---|---|---|---|
| Jeremy | 1st Nodal | 14.45 | 13.52 | |
| Bridget | 1st Antinodal | 21.64 | 23.42 | |
| Jane | 3rd Antinodal | 16.22 | 21.67 | |
| Jose | 4th Nodal | 25.22 | 19.10 |
Audio Guided Solution
This problem pertains to a two-point source interference pattern. Two-point source interference patterns are created by two sources of waves which produce waves emanating out through the space surrounding the sources and interfere in such a way that you get a collection of nodes and antinodes in an alternating pattern. Further information about two-point source interference patterns can be found if you click the link back to the physics classroom tutorial. There you'll find the background information and equations which relate the key variables in the pattern. Here in this question, we're told that Jeremy is standing up on a nodal line. Being on a nodal line, then the difference in distance traveled from one of the speakers to that location compared to the other speaker at that location is equal to a half number of wavelengths. So we can find the difference in distance traveled from the speakers to the point where Jeremy is standing. It's just 14.45 minus 13.52 meters. That would come out to be a value of 0.93 meters. Now this is equal to one-half of a wavelength, so if we divide 0.93 by half, we get a wavelength of 1.86 meters. The same process can be done for all the students who are standing on either an antinodal or nodal line. It begins by first finding the so-called path difference or difference in distance traveled from the speaker out to that point. So for Bridget, it would be 23.42 minus the 21.64. That would give me a difference in distance traveled of 1.78 meters, and being that she's standing on the first antinodal line with an antinodal line number of one, we would say that this 1.78 meters is equal to one times wavelength. Now for Jane, she's standing on the third antinodal line and the difference in distance traveled is 21.67 minus 16.22. That comes out to be 5.45 meters, and she's standing on the third antinodal line, and so this 5.45 meters is equal to three wavelengths. We can divide the 5.45 meters by three, and we get 1.8167 meters. We can round that to the second decimal place. Finally, we have for Jose that he's standing on the fourth nodal line where the difference in distance traveled can be found as 25.22 minus the 19.10. This gives me a difference in distance traveled or a path difference of 6.12 meters, and that, since it's the third, fourth nodal line, is equal to 3.5 wavelengths. So if I divide the 6.12 meters by 3.5, I get 1.7486 meters, and I can round that to the second decimal place.
Solution
| Student | Nodal or Antinodal Line | Distance to Speaker 1 (m) | Distance to Speaker 2 (m) | Wavelength (m) |
|---|---|---|---|---|
| Jeremy | 1st Nodal | 14.45 | 13.52 | 1.86 |
| Bridget | 1st Antinodal | 21.64 | 23.42 | 1.78 |
| Jane | 3rd Antinodal | 16.22 | 21.67 | 1.82 |
| Jose | 4th Nodal | 25.22 | 19.10 | 1.75 |
Habbits of an Effective Problem Solver
- Read the problem carefully and develop a mental picture of the physical situation. If necessary, sketch a simple diagram of the physical situation to help you visualize it.
- Identify the known and unknown quantities and record in an organized manner, often times they can be recorded on the diagram itself. Equate given values to the symbols used to represent the corresponding quantity (e.g., \(\descriptive{v}{v,velocity} = \num{3e8}\unit{\meter\per\second}\), \(\descriptive{λ}{λ,wavelength} = 554 \unit{\nano\meter}\), \(\descriptive{f}{f,frequency} = \colorbox{gray}{Unknown}\)).
- Use physics formulas and conceptual reasoning to plot a strategy for solving for the unknown quantity.
- Identify the appropriate formula(s) to use.
- Perform substitutions and algebraic manipulations in order to solve for the unknown quantity.
Read About It!
Get more information on the topic of Light Waves and Colors at The Physics Classroom Tutorial.