15. Interference due to Path Difference.

Interference between two waves often occurs due to a difference in the length of the path that two waves travel. In this simulation notice that the top two waves start out in phase (their maximums occur at the same time) but the top wave travels further than the other wave. The bottom graph shows the sum of the two waves as if they arrived at the same location at the same time. Depending on the path difference the two waves may arrive exactly in phase (leading to constructive interference), exactly out of phase (destructive interference) or something in between.


Note: Animations may take a few seconds to load.


Questions:

15.1. Look at the three cases of a path difference equal to one wavelength, a path difference of half a wavelength, and a path difference of 3/4 a wavelength. Describe what happens in each case.

15.2. Which case has the largest combined amplitude? Explain.

15.4. Imagine this simulation represents monochromatic (single wavelength) light reflecting off a surface with two levels. Only the reflected waves (going to the right) are show (the incoming waves coming from the right are not shown). In which case would there be no reflection from the surface?

15.5. Suppose you wanted to make a "stealth" jet plane which was non-reflective to a particular wavelength of radar. Describe how you might try to do this by modifying the surface of the plane.

The following buttons simulate three different wavelengths with the same path difference for each. (Note: The simulation is not to scale but rather is only suggestive.)

15.6. Which color experiences total destructive interference?

15.7. Suppose all three colors were reflected off the same two level surface. Which color would experience constructive interference and thus be most strongly reflected?


Path differences can occur in a number of different ways. In the Ripple Tank simulation of the double slit experiment (simulation 10.4) the distance to a point on the screen is different for each source so the light experiences a path difference. If the path difference is a whole number of wavelengths (1, 2, 3...) there is constructive interference and a bright spot for that color appears on the screen. If the path difference is 1/2, 3/2, 5/2 ..etc. of a wavelength then a dark spot appears due to destructive interference.

Path differences can also occur due to reflection from a surface that has multiple layers. In this case the waves that hit the deeper surface travel twice the distance between the surfaces before recombining (2D in the simulations above for a wave that comes in from the right and reflects back to the right).

15.8. A music CD has information stored on it in the form of tiny divots blasted into the surface with a laser. Explain why you only see one particular color when looking at a CD at a fixed angle.

15.9. Some insect wings and the feathers of some birds (for example peacocks) exhibit a feature known as iridescence. From a fixed angle only one color of reflected light can be seen. Explain this phenomena given the fact that insect wings and feathers consist of overlapping layers causing the surface to be multi layered.


The formula for constructive interference due to a path difference is given by  where n is the index of refraction, l is the wavelength d is the path difference and m is a number. In the simulation above  d = 2D is the actual path difference; a wave reflected off the upper surface must travel an extra distance of 2D to catch up with a wave reflected off the lower surface.

15.10. Soap bubbles show different colors at different places on the bubble. So do oil slicks on water. In both cases light reflects off the upper and lower surfaces of the layer of soap or oil. Explain the different colors in terms of path difference (Hint: draw a picture where the wall of the soap bubble is nearly the same thickness as one wavelength and explain why the path difference is twice the thickness of the soap).


There is one other detail needed to explain the soap bubble and oil slick color phenomena completely. The light reflecting from the top surface is going from a "soft" medium (air) to a "stiff" medium (soap) but the light reflecting from the bottom layer of the soap is going from a "stiff" medium (soap) to a "soft" medium (air inside the bubble).

15.11. From what you learned about reflection and transmission of waves from "stiff" and "soft" boundaries in simulation 13 and 13, what happens to a wave reflected from the top layer of a soap bubble which does not occur at the bottom layer?

15.12. If the path difference, 2D, for a particular thickness of soap film was just right for destructive interference but there was a 180o phase change for the top reflected wave but not the bottom reflected wave, what would happen?
Credits.
Go To: IUS Physics Top Page.
Contact Dr. K. Forinash, for comments/suggestions/corrections.