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P147. Superposition and Interference

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.

  1. 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?

  2. 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.

  3. Scan and upload your superposition plot.

Part B. Interference of point sources

  1. 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.
Figure A Figure B
Figure C Figure D
  1. 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.
Figure E Figure F

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. 

  1. For point R, determine the ratio PD/λ. You can determine this without measurement by examining the geometry of the figure.

  2. Tell how you know that point R is a point of constructive interference.
Figure G
  1. 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.

    1. 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.

    2. Using a centimeter ruler, measure PS1 and PS2 to the nearest 0.01 cm.

    3. Calculate the path difference PS2 - PS1.

    4. 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.

    5. Calculate the ratio of the path difference from part c to the wavelength that you measured in part d.

    6. Estimate the uncertainty in your calculated ratio in part e. Show your work. See Lab FAQ to review how to do this.

    7. To within the uncertainty of your ratio is the interference at Point P constructive or destructive? Explain.

  2. 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.

  3. 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 your ratio.

Figure H
 


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