state and explain interference of light and find the condition for constructive and destructive interference also find its expression
Answers
Answer:
In the last section we discussed the fact that waves can move through each other, which means that they can be in the same place at the same time. This is very different from solid objects. Thus, we need to know how to handle this situation. As it turns out, when waves are at the same place at the same time, the amplitudes of the waves simply add together and this is really all we need to know! However, the consequences of this are profound and sometimes startling.
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If we stand in front of the two speakers, we will hear a tone louder than the individual speakers would produce. The two waves are in phase. Now imagine that we start moving on of the speakers back:
At some point, the two waves will be out of phase � that is, the peaks of one line up with the valleys of the other creating the conditions for destructive interference. If we stand in front of the speakers right now, we will not hear anything! This must be experienced to really appreciate. Equally as strange, if you now block one speaker, the destructive interference goes away and you hear the unblocked speaker. In other words, the sound gets louder as you block one speaker!
How far back must we move the speaker to go from constructive to destructive interference? We know that the distance between peaks in a wave is equal to the wavelength. If we look back at the first two figures in this section, we see that the waves are shifted by half of a wavelength. So, in the example with the speakers, we must move the speaker back by one half of a wavelength.
What happens if we keep moving the speaker back? At some point the peaks of the two waves will again line up:
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The proper way to define the conditions for having constructive or destructive interference requires knowing the distance from the observation point to the source of each of the two waves. Since there must be two waves for interference to occur, there are also two distances involved, R1 and R2. For two waves traveling in the same direction, these two distances are as follows:
Destructive interference:
Once we have the condition for constructive interference, destructive interference is a straightforward extension. The basic requirement for destructive interference is that the two waves are shifted by half a wavelength. This means that the path difference for the two waves must be: R1 � R2 = l /2. But, since we can always shift a wave by one full wavelength, the full condition for destructive interference becomes:
R1 � R2 = l /2 + nl .
Now that we have mathematical statements for the requirements for constructive and destructive interference, we can apply them to a new situation and see what happens.
Waves with the same frequency traveling in opposite directions.
To create two waves traveling in opposite directions, we can take our two speakers and point them at each other, as shown in the figure above. We again want to find the conditions for constructive and destructive interference. As we have seen, the simplest way to get constructive interference is for the distance from the observer to each source to be equal. Using our mathematical terminology, we want R1 � R2 = 0, or R1 = R2. Looking at the figure above, we see that the point where the two paths are equal is exactly midway between the two speakers (the point M in the figure). At this point, there will be constructive interference, and the sound will be strong.
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What we see is a repeating pattern of constructive and destructive interference, and it takes a distance of l /4 to get from one to the other. Where have we seen this pattern before? At a point of constructive interference, the amplitude of the wave is large and this is just like an antinode. At a point of destructive interference, the amplitude is zero and this is like an node. So, if we think of the point above as antinodes and nodes, we see that we have exactly the same pattern of nodes and antinodes as in a standing wave. From this, we must conclude that two waves traveling in opposite directions create a standing wave with the same frequency! You can get a more intuitive understanding of this by looking at the Physlet entitled Superposition.
Translating the interference conditions into mathematical statements is an essential part of physics and can be quite difficult at first. Moreover, a rather subtle distinction was made that you might not have noticed. On the one hand, we have some physical situation or geometry. This refers to the placement of the speakers and the position of the observer. This really has nothing to do with waves and it simply depends on how the problem was set up. Given a particular setup, you can always figure out the path length from the observer to the two sources of the waves that are going to interference and hence you can also find the path difference R1 � R2.