II Two radio antennas are separated by 2.0 m. Both...

Key Concepts

Path Difference and Phase Relationship

The phase relationship between waves depends on the difference in the distance each wave travels before combining. This path difference is key in determining the conditions under which constructive (or destructive) interference occurs, making it a crucial concept for analyzing interference patterns.

Wavelength Determination

Wavelength is the distance between corresponding points on consecutive cycles of a wave, and it is calculated by dividing the speed of the wave by its frequency. This relationship is essential in linking the frequency of a wave to its spatial properties in interference and diffraction phenomena.

Wave Interference

Wave interference is the phenomenon that occurs when two or more waves meet and superpose, resulting in a new wave pattern. This concept is fundamental in understanding how the overlapping of waves from different sources leads to regions of increased or decreased amplitude in a medium.

Constructive Interference

Constructive interference happens when waves arrive in phase, meaning their peaks and troughs align, so their amplitudes add together, resulting in maximum intensity. The condition for this is that the path difference between the waves is an integer multiple of the wavelength.

Transcript

00:01 All right.

00:04 So i wrote down the pertinent information that the distance between the two speakers is two meters.

00:11 That means that as i indicated the speakers in green here, there's the locations of the two speakers.

00:18 They are each one meter from the center.

00:21 I also listed that the frequency is 750 gigahertz.

00:26 The speed of light is three times 10 to the eighth meters per second.

00:32 And i know for constructive interference that the difference in the path length has to be a multiple of lambda.

00:44 Now, if the person who hears it, the observer or whatever, walks around the circle, then i know he'll be making a path of x squared plus y squared equals 10 squared, because that's the equation of the circle.

01:02 But if i take the same thing and i think about this farther path, the farther blue path, well, that's going to be, the length of that is going to be the square root of x squared plus y squared.

01:24 But y is a little bit longer.

01:27 It's y plus one.

01:31 It's one meter longer.

01:33 And then if i subtract off the shorter path, the shorter path is x squared plus y minus one squared.

01:50 And i want to do this for every, and i'm going to write that down, for every point on the circle, such that y is going from this equation, y is the square root of 100.

02:12 Minus x squared, 10 squared minus x squared.

02:16 That'll give us y.

02:17 So now i'm going to substitute y back into the equation.

02:23 So this is going to give me x squared plus y.

02:28 Well, y is the square root of 100 minus x squared, y plus one squared, minus the square root of x squared, minus the square root of x squared, plus y minus 1.

02:48 So that's going to be the square root of 100 minus x squared, this time minus 1 squared.

02:59 And that is going to be the difference in the path length, which i want to be lambda m, as i indicated above right here.

03:10 I want it to be lambda m for constructive interference.

03:15 And lambda is v over f.

03:18 So i want it to be v over f, m.

03:23 So if i solve this for m, i get m is f over v, the square root of x squared plus the square root of 100 minus x squared plus 1 squared minus the square root of x squared plus the square root of 100 minus x squared minus 1 squared.

03:59 All right, i thought about solving this algebraically, and i don't think i really want to try that.

04:07 So what i did is i put it into a graph.

04:10 I've also shared my screen so that you can see the graph.

04:15 I put this into the graph and it gives me this.

04:19 You can see that i put in all of the numbers correctly and it gives me this circle basically...

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