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In the figure below, two speakers, $\mathrm{S}_{1}$ and $\mathrm{S}_{2}$, emit sound waves of wavelength 2 m, in phase with each other.IMAGE IS NOT AVAILABLE TO COPYLet $A_{p}$ be the amplitude of the resulting wave at Point $P,$ and $A_{Q}$ the amplitude of the resultant wave at Point $Q .$ How does $A_{p}$ compare to $A_{Q} ?$(A) $A_{\mathrm{P}} < A_{\mathrm{Q}}$(B) $A_{\mathrm{P}}=A_{\mathrm{Q}}$(C) $A_{\mathrm{P}} > A_{\mathrm{Q}}$(D) $A_{\mathrm{P}} < \mathrm{o}, A_{\mathrm{Q}}>0$

Physics 101 Mechanics

Physics 102 Electricity and Magnetism

Chapter 9

Waves

Sound and Hearing

Mechanical Waves

Electromagnetic Waves

Rutgers, The State University of New Jersey

University of Washington

Hope College

McMaster University

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Yeah, in this example we are going to look at a situation in which waves are interfering in space. Um And although the example works with sound waves, the idea is the same, no matter what type of wave you have is that if waves are in phase when they get together at a point. Um So we're looking at waves that have the same wavelength. Uh Sometimes those are called monochromatic late waves, especially if it's light, but they have to have the same wavelength in order to get a coherent pattern. Um But what governs how they interfere is a path length Difference between two waves. So two waves that come together in space um Wave one, wave two, if they start out in phase and they stay in phase, you get what's called constructive interference if they get out of phase um with uh huh half of a wave length difference between them, so that the crest of one wave adds to the trough of another, You get 180° out of phase. And that's called destructive interference. And the condition for these two situations if they both start out together um from say the same source or two sources that have been carefully uh programmed to be in phase. Um The condition for constructive interference is a difference in path length in space equal to an integer number of full wavelengths. So and could be zero. Starting out at the source together. Could be 1, 2 et cetera. But they have to travel through space to get together again with an integer multiple of wavelengths difference in how they got there. Difference in path length for destructive interference, on the other hand, is an integer multiple of half wavelengths. So sometimes this is written as n plus one lambda over two. Where and again can be zero um two for etcetera. So if you you put in those even ends, um different ways to write it. But um if you put in even ends, uh then you wind up with uh say one half lambda path difference or three halfs lambda et cetera. Um So what matters is the difference in path link between two waves. So let's take as an example, let's pose that you have to sound generators will calm S. One and S. Two. And let's say they have been carefully programmed to send out monochromatic sound waves. Both have λ Equal two m. I don't know offhand whether those are something that you could hear, what you have to work that out, but they're separated by three m. Down the middle is 1.5 m. But they're full separation is three m. And you have um a sound detector that you can move around, but we'll pretend that they also start out programmed in phase with each other at the source. So they have been fiddled with so that they're putting out nice pretty sound waves do. Um I don't know that's the frequency. But anyway they're putting out nice um Pretty sound waves that oscillate with a clean amplitude and period and wavelength. Um And we've got a detector that we could put at either point P. Or point Q. And point P. Is four m away. So here's our detector four m away from source one. Um But it's five m away that right triangle situation five m away from source to and we could ask ourselves what would we hear at point P. So that's the question. Um And what we would have happening there is um Source one has traveled a path length of two full wavelengths. Two times two m is four m. Source too will have traveled 2.5 wavelengths Over that distance of five m, so five m divided by two m, gives us 2.5 way and this and notice that those are one half of a wavelength now out of phase. Um So they are one half lambda out of phase. And that means the path links have allowed them to shift into a situation more like the second one that I showed, and so they will actually cancel each other out with destructive interference. Um Meanwhile, if you moved your detector down to the central position here, um Each of those would have traveled basically five m, Well not quite five m, but um and I'm not even sure we need to figure that out with a pythagorean theorem. But what we see in um detector Q is that the two sources have traveled identical path lengths, identical paths through space, same length, um which means that they come together in constructive interference. And so you'd hear loud sound at point Q, and you would hear a quiet sound at point P. Or the amplitude of the resulting sound wave at Q should be much much greater than the amplitude at point P. Yeah.

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