Interference
37 Interference in two dimensions
Interference in two dimensions
So far, this chapter has only examined interference along a straight line (to keep things simple). Real waves- like water waves, sound and light- exist in more than just one dimension. What does interference look like in two-dimensional world (like the surface of a lake) or in the three-dimensional world we experience light and sound in?
This section explores the interference pattern produced by two sources. The example is shown in two dimensions, but the math also applies to the real three-dimensional world. To see what interference looks like, check out this stunning photo of a bee making waves [1] on the surface of a pond. (Due to copyright restrictions, I can only provide the link to the picture).
Two point sources
A point source is the simplest source of waves. Point sources (also called monopoles) send out circular waves (in two dimensions) or spherical waves (in three dimensions). What happens when two identical point sources make sound waves? The two sources have the same frequency, same amplitude and are in phase. Will there be interference? If so, where will there be nodes and antinodes? How can we figure out where those will be?
It might seem (at first glance) that there shouldn’t be any destructive interference anywhere- after all, both sources are making identical waves at the same time. But there’s a catch: sound (or any wave, for that matter) takes time to get from the source to the detector. Both speakers emit a short burst of sound at the same time. Will you hear both sounds at the same time? The answer is “no” (unless you happen to be sitting the same distance from both speakers). The sound from the nearer speaker will arrive a split second earlier than sound from the farther speaker. Both sounds start the journey at the same moment and travel at the same speed, but the sound from the far speaker has a slightly longer trip. This extra distance creates the time delay between the arrival of the two sounds at the listener.
This extra distance is important for interference. For example, the extra distance is half a wavelength, sound from the near speaker arrives half a cycle earlier than the sounds from the far speaker. As a crest made by the near speaker arrives at the listener, it meets the trough that was made by the far speaker half a cycle earlier. The sounds interfere destructively and the listener hears nothing. This destructive interference occurs at all locations that are half a wavelength closer to one speaker than the other. Destructive interference will also occur if the extra distance is [latex]1.5\lambda[/latex], [latex]2.5\lambda[/latex], [latex]3.5\lambda[/latex], etc. Each nodal line corresponds to a different amount of “extra distance.” The result is a nodal line– a line along which the medium doesn’t move. On a pond, nodal lines are visible as lines of still water- check out the “pond scene” about 4:25 into Veritasium’s youTube video “T