Write your solutions as directed.
You'll need to have read section 14.7 before doing
these problems. Submit your work to the corresponding WebAssign assessment.
Part A. Superposition
The purpose of this problem is to provide practice in plotting the superposition of two waves. Print this page in landscape orientation.
Consider the graph to be a snapshot (at an instant of
time) of two waves on the same medium. Let's assume the waves are moving at
a speed of 20 m/s. What is the frequency of each wave?
Plot by hand the superposition of the two waves point-by-point
on the graph. This amounts to adding the y-coordinates of each wave at particular values of x. Mathematically speaking, the y-coordinate of the superposition as a function of x is ysuper(x) = yA(x) + yB(x). In order to get a smooth curve for the superposition, plot
ysuper for values of x in increments of 0.05 m. When you've plotted all the points, draw a smooth curve through them. This is the shape of the wave that you would actually see on the medium.
- Scan and upload your superposition plot.
Part B. Interference of point sources
- The figures below represent two interfering
sources of sound. The sources have the same wavelength, frequency, and
phase. In order to help distinguish waves from the two sources, they
are color coded red and green. The red and green circles represent the
crests of the waves; therefore, the perpendicular distance between the
crests is the wavelength. Regions where the crests overlap are areas
of constructive interference. These appear as fuzzy bluish rays extending
from the sources. (Click on Figure A for help in identifying the lines
of constructive interference.) Examine the 4 figures. What are the variables?
That is, what is changing from one diagram to the next? Here's part of
the answer: The dependent variable is the angular spacing of the lines
of constructive interference. What is the independent variable?
State in a sentence the relationship between the variables.
- Now examine the following two figures. The
independent variable is different than before, but the
dependent variable is the same as for the figures in problem 1. What
is the independent variable? State in a sentence the relationship
between the variables.
Examine Figure G below. It also shows two
interfering sources. These are labeled S1 and S2.
Three points, P, Q, and R are labeled. Q and R are points of
constructive interference; wave crests overlap here. P is a point of
destructive interference where crest and trough overlap. The key
concept in determining whether waves interfere constructively or
destructively is the path difference. This is the difference in
distance that waves must travel from the sources to the point in question.
We'll call the path difference PD for short. When the PD is an
integral number of wavelengths, the interference is constructive. When
the path difference is an odd-integer multiple of half wavelengths, the
interference is destructive. These are important relationships to
master now, as we'll see them again when we study light. In symbolic
notation, the conditions are:
Constructive interference: PD = n(wavelength),
where n = 0, 1,2,... (any non-negative integer)
Destructive interference: PD = (n + ½)(wavelength),
where n = 0,1,2....(any non-negative integer)
In Figure G, note that the path difference for point Q is
S1Q - S2Q = 0. Therefore, this is a point of
constructive interference. Now consider Point P. Were you to measure the difference S2P
- S1P, you would find that the result was half a wavelength. Therefore, P is a point of destructive interference.
For point R, determine the ratio PD/λ.
You can determine this without measurement by examining the geometry of the
- Tell how you know that point R is a point of constructive interference.
See Figure H below. Points Q and R are points of constructive interference as shown in the
previous problem. Do the following. Pay close attention to measurement
precision and accuracy and significant figures in calculations.
Click here for a larger version of Figure H. Print the diagram. On that page, measure across the diagonal AB of
the box surrounding the figure. Measure to the nearest 0.01 cm. This value will be used to calculate a
scale factor for checking your work.
Using a centimeter ruler,
measure PS1 and PS2 to the nearest 0.01 cm.
Calculate the path difference
PS2 - PS1.
Devise a method that you can use to
measure the wavelength directly from the printout to ensure an
accuracy to within 0.001 cm in your value of
wavelength. Use your method and
record your result.
Calculate the ratio of the path
difference from part c to the wavelength that you measured in part
uncertainty in your calculated ratio in part e. Show your work. See Lab FAQ to
review how to do this.
To within the uncertainty of your
ratio is the interference at Point P constructive or destructive?
Describe the method that
you used to obtain an accuracy of 0.001 cm in the wavelength that you
measured in 5d. Explain why your method
gives that accuracy.
Now repeat some of the steps
in item 5 for point T. Using a centimeter ruler,
measure TS1 and TS2 to the nearest 0.01 cm.
Then subtract the values to obtain the path difference. Calculate the ratio
of the path difference to the wavelength and determine whether the
interference is constructive or destructive to within the uncertainty of