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$\bullet$ $\bullet$ Suppose that the situation is the same as in the previousproblem, except that the two speakers are $180^{\circ}$ out of phase.Repeat parts $(a),(b),$ and $(c)$ of that problem.

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Physics 102 Electricity and Magnetism

Physics 103

Chapter 26

Interference and Diffraction

Electromagnetic Waves

Reflection and Refraction of Light

Rutgers, The State University of New Jersey

University of Sheffield

University of Winnipeg

McMaster University

Lectures

02:30

In optics, ray optics is a geometric optics method that uses ray tracing to model the propagation of light through an optical system. As in all geometric optics methods, the ray optics model assumes that light travels in straight lines and that the index of refraction of the optical material remains constant throughout the system.

10:00

In optics, reflection is the change in direction of a wavefront at an interface between two different media so that the wavefront returns into the medium from which it originated. Common examples include the reflection of light, sound and water waves. The law of reflection says that for specular reflection the angle at which the wave is incident on the surface equals the angle at which it is reflected. Reflection may also be referred to as "mirror image" or "specular reflection". Refraction is the change in direction of a wave due to a change in its speed. The refractive index of a material is a measure of its ability to change the direction of a wave. A material with a higher refractive index will change the direction of a wave to a greater degree than a material with a lower refractive index. When a wave crosses the boundary between two materials with different refractive indices, part of the wave is refracted; that is, it changes direction. The ratio of the speeds of propagation of the two waves determines the angle of refraction, which is the angle between the direction of the incident and the refractive rays.

03:24

I Suppose that the situati…

01:17

Two in ~phase loudspeakers…

04:26

(II) The two sources of so…

01:35

01:21

If the loudspeakers in Pr…

If the loudspeakers in Pro…

01:03

As the first speaker is ad…

04:01

Two identical speakers (1 …

05:58

The two sources of sound i…

06:49

Two loudspeakers, $A$ and …

02:18

Review Example 1 in the te…

14:52

Suppose that the two speak…

02:21

$\bullet$$\bullet$ Two ide…

06:33

05:17

Loudspeakers A and B are v…

already, So this one has a little bit trig ear than the previous problems we've been trying. So we have two speakers again and be separated by 15 meters, and there's a person standing in the middle. Then, at some point, our person decides to start walking towards one of the speakers, and we want to know at what point are they going to hear constructive or destructive interference? So when are they going to hear the sound the loudest as they walk along? And when are they going to hear the sound go away as they walk along? The important part to notice about this one is that our speakers are out of phase 580 degrees. So if we draw that with our waves, that's like the wave from one speaker starts off going up, down, up, down. Then the other one is going to start the exact opposite. Down, up, down, up. So in order to be constructive or destructive, that's some observation point. You either need to add another half a wavelength. You tow one so that they're both ending like that to get constructive interference or, as it was already drawn with your peak and your trough match each other. That's when you're gonna get destructive. So this means that the equations that we've been using for constructive and destructive interference are gonna be switched because of this out of phase by 180 degrees. So when destructive interference is gonna happen and that's when our path links, we're going to be interred Her multiples of the wavelength because the 2nd 1 is already shifted by that half into jerk from this out of phase, likewise are constructive interference is gonna happen when you add in the additional shift by half a wavelength like we did here. So that's a conceptually tricky part about this problem. So if you wrap your head around that, then we're good to go. So let's go ahead and souls for when this happens. Well, you draw my person again. So you have speaker, eh speaker be. These are 15 meters apart and my person is starting right smack dab in the middle, and then they're deciding to walk some distance. So if we assume that they work a little bit until they hear, we'll say, we'll do constructive first, some constructive interference. So now they're over here. So the past between speaker and my person over here, it's gonna be my halfway point. So seven and 1/2 meters to get to here, plus some unknown value X so say are two equals seven when 1/2 meters plus six. Likewise between B and this person we'll switch to green where my starting value seven and 1/2 meters away minus this distance X So if I go into my handy dandy equation are two minus are one Remember, in this case for constructive needs to be offset by half the wavelength times lambda so are 27 and 1/2 meters plus X minus seven and 1/2 meters minus six You gotta end plus 1/2 times blamed it. So now I have my seven and 1/2 meters minus seven and 1/2 meters. So that was cancelled up the x minus of minus X. So it's going to give me two X that's equal to M plus 1/2 Hans Landa. All right, so now Nix is equal to divide both sides by two. So you get in plus 1/2 lambda over too. So the first time that this is gonna happen is when M is equal to zero. That's gonna give me Lambda over four. We're in a pause here for a second before we plug in our numbers and go to the next part so we can do all of our calculations at the same time. All right, so for part B, same thing are too most seven and 1/2 meters plus x are one the seven and 1/2 meters Minor sex. When our interference doesn't happen at integer values of Lambda because they're out of phase, so am I. Seven and 1/2 meters plus x minus seven and 1/2 Meters minus six is equal to m lambda again. My seven and 1/2 meters Cancel out X minus A minus X is to fix nickels and lambda So in M equals zero. That's when the person is standing in the middle. They haven't moved yet, so we're gonna use em equals one as the first time that they hear that interference pattern. Listen, you lambda well, her too. All right. It's now for a constructive. We had X was gonna be Landa over four and for destructive necks. We're gonna be Landover too. Well, What's Lando? They didn't tell us what the wavelength was. They told us what the frequency was and what the velocity of sound was. So you have to do one extra step. So Lamda is related to those two quantities with this relationship. So it's the velocity of the wave in the medium divided by the frequency of that wave. So in this case, they said that our velocity was 340 meters per second, just the typical value for the speed of sound and air divided by the frequency which was 250 hertz. Now Hertz is units of one over seconds. So this is going to give us a value of 1.36 meters. So now you can come over here, plug these in. So 1.36 meters over four is going to give me 0.34 meters. We're a 34 centimeters in Landover to 1.36 meters, divided by two going to give me a 0.68 meters or 68 centimeters.

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