Two in-phase loudspeakers are located at (x, y) coordinates...

Key Concepts

Wave Interference

Wave interference occurs when two or more waves combine to produce a new wave pattern. The interference can be constructive, where the waves reinforce each other, or destructive, where they cancel each other out, depending on their relative phases.

Wavelength

Wavelength is the spatial period of a periodic wave, directly related to the speed and frequency of the wave. It plays a crucial role in determining the interference pattern, as the phase difference depends on the ratio of the path difference to the wavelength.

Constructive and Destructive Interference

Constructive interference takes place when the phase difference is a multiple of 2?, leading to an increased amplitude, while destructive interference occurs when the phase difference is an odd multiple of ?, reducing or cancelling the amplitude. These concepts are key in predicting the interference pattern in any multi-source wave system.

Superposition Principle

This principle states that when two or more waves overlap in space, the resultant wave is the sum of the individual waves. It is fundamental in understanding how different sound waves interact, leading to regions of increased or decreased amplitude.

Phase Difference and Path Difference

The phase difference between waves is often caused by differences in the path lengths they travel. Even if sources are in phase, variations in distance to a given point result in a phase shift, which determines the type and extent of interference at that point.

Transcript

00:01 All right, i've drawn a diagram of this situation.

00:04 Notice that i have two speakers that are four meters apart.

00:08 That's because they go from y equals negative 2 to y equals positive 2.

00:14 There's also three meters between the speakers and the points in question.

00:21 I drew r sub a, which would be, and i'd rather use a capital a here, the distance to point a.

00:37 And notice that that is, and you know what, i'm going to also write down 1 and 2, so i'm going to call these points 1 and 0 .2.

00:46 The distance from point 1 to point a is d squared plus r1 squared, where r1 is 3.

00:59 Okay? the distance from point 2 to point a, is also d squared plus r1 squared.

01:16 And so the distances are the same.

01:18 So at point a, when we write down that the amplitude is 2a, cosine of one -half delta theta, where delta theta is 2 pi delta r over lambda plus delta phi phi sub -zero, the phase shift is zero.

01:37 Delta r is also zero and the cosine of zero therefore is is one and so the amplitude at point a so i don't write that there i'm going to write amplitude at point a maybe i'll write it over here now is 2a all right the amplitude of point b, so if we're going to do point b, it is half a meter upwards.

02:26 So that means that the y value is going to be closer to point two, and it's going to be farther away from point one.

02:33 So, r from two, whoops, two to b is going to be the square root of d minus, 1 1� squared plus r1 squared.

03:12 R1b is going to be the square root of d plus 1 1⁄2.

03:26 So delta r2b is going to be r1b, actually it's going to be r1b, minus r2b, because r1b is going to be greater than r2b.

03:50 Okay.

03:53 Now, delta phi is going to be 2 pi, r1b minus r2b over lambda.

04:24 Okay.

04:27 And so the amplitude is going to be 2 times a times the cosine of half of delta 5.

04:46 Well, half of delta phi is just going to be pi over lambda times r1b minus r2b.

04:59 Okay, so i can put this part in a graphing calculator or in a calculator and calculate what it comes up with.

05:12 But i want to do something else here.

05:17 For c, all i have to do, is multiply, put a multiply by two in here, two times and two times.

05:31 So for c, i just need to do two times.

05:34 For d, i just need to do three times.

05:38 For e, do we have an e? for e, i just have to do four times away.

05:47 Because let's say for e, four times one half is two.

05:52 So if i just count this off, one half, one and a half, two.

06:01 So i can make this easier by just putting that factor into the calculator and i'm going to use desmos .com.

06:11 So i'm going to type in two times the cosine of pi over lambda...

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