A Simple Introduction to Interference
Learning Goal: To understand the basic principles underlying interference.
One of the most important properties of waves is the principle of superposition. The principle of superposition for waves states that when two waves occupy the same point, their effect on the medium adds algebraically. So, if two waves would individually have the effect "+1" on a specific point in the medium, then when they are both at that point the effect on the medium is "+2." If a third wave with effect "-2" happens also to be at that point, then the total effect on the medium is zero. This idea of waves adding their effects, or canceling each other's effects, is the source of interference.

First, consider two wave pulses on a string, approaching each other. Assume that each moves with speed meter per second. The figure shows the string at time . The effect of each wave pulse on the string (which is the medium for these wave pulses) is to displace it up or down. The pulses have the same shape, except for their orientation. Assume that each pulse displaces the string a maximum of meters, and that the scale on the x axis is in meters.

Part A
At time , what will be the displacement at point ?
 = 0
Part B
Choose the picture that most closely represents what the rope will actually look like at time .
 ABCD
The same process of superposition is at work when we talk about continuous waves instead of wave pulses. Consider a sinusoidal wave as in the figure.
Part C
How far to the left would the original sinusoidal wave have to be shifted to give a wave that would completely cancel the original? The variable in the picture denotes the wavelength of the wave.
 =
Part D
In talking about interference, particularly with light, you will most likely speak in terms of phase differences, as well as wavelength differences. In the mathematical description of a sine wave, the phase corresponds to the argument of the sine function. For example, in the function , the value of at a particular point is the phase of the wave at that point. Recall that in radians a full cycle (or a full circle) corresponds to radians. How many radians would the shift of half a wavelength from the previous part correspond to?
Part E
The phase difference of radians that you found in the previous part provides a criterion for destructive interference. What phase difference corresponds to completely constructive interference (i.e., the original wave and the shifted wave coincide at all points)?
Part F

Since sinusoidal waves are cyclical, a particular phase difference between two waves is identical to that phase difference plus a cycle. For example, if two waves have a phase difference of , the interference effects would be the same as if the two waves had a phase difference of . The complete criterion for constructive interference between two waves is therefore written as follows:

Write the full criterion for destructive interference between two waves.

 phase difference =
The phase for a plane wave is a somewhat complicated expression that depends on both position and time. For most interference problems, you will work at a specific time and with coherent light sources, so that only geometric considerations are relevant. Consider two light rays propagating from point A to point B in the figure, which are apart. One ray follows a straight path, and the other travels at a angle to that path and then reflects off a plane surface to point B. Both rays have wavelength .
Part G
Find the phase difference between these two rays at point B.
Part G.1
Find the difference in distance
Find the difference in length between the direct path and the reflected path. You can use the fact that triangle ABC is an equilateral triangle.