Two speakers A and B are 3.50 m apart, and each one is emitting a frequency of 444 Hz. However,

Two speakers A and B are 3.50 m apart, and each one is...

Key Concepts

Wavelength

Wavelength is the distance between successive points of equal phase in a wave, such as two consecutive peaks. It is determined by the speed of the wave and its frequency. In scenarios involving sound, the wavelength is crucial for understanding interference patterns as it sets the scale for constructive and destructive interference conditions.

Wave Interference

This concept explains how overlapping waves from two or more sources combine to form a new wave pattern. Depending on the phase and amplitude of the individual waves, the interference can be constructive, where the amplitudes add up, or destructive, where they cancel each other out. In the context of sound, the interference of waves produced by two speakers leads to areas of increased and decreased sound intensity.

Phase Difference

Phase difference refers to the offset between the peaks (and troughs) of two waves of the same frequency. A phase difference can be introduced by varying the emission time or by inherent signal delays. This alters how the waves superimpose on each other, directly influencing whether they interfere constructively or destructively at a given point.

Path Difference

Path difference is the difference in the distance traveled by waves from their respective sources to a particular observation point. Even with no initial phase difference, different distances can cause the waves to arrive shifted in phase. When combined with any initial phase shift, the path difference determines the nature of the interference observed at that location.

Far-field Approximation

The far-field approximation assumes that the observation point is much farther from the source compared to the separation between the sources. This simplifies the geometry of wave propagation, allowing the interference pattern to be analyzed based primarily on angles relative to the centerline rather than complex distance calculations.

Transcript

00:01 In this question, we are going to find angular locations far away from a pair of speakers who are emitting the same frequency tone but are a quarter of a cycle out of phase.

00:16 We want to know where they have destructive interference.

00:23 So this is going to be a little bit different than what you've done with destructive interference in the past, where you didn't have to worry about a phase difference and your sound sources were in phase, and so destructive interference only meant needing half a wavelength extra in your half length difference.

00:54 So let's pretend that this line i'm about to draw is far away from our speakers.

01:10 Right, this is far away, but it's close enough i can actually sketch some things.

01:19 So notice that we're asked to find angles on both sides of the center line, because this is actually not going to be symmetric because of that quarter cycle difference in how the sounds are being emitted.

01:34 So for a above the center line, i'm gonna, you know what i'm gonna do, i am actually gonna let a be a little bit more angled.

01:54 And again, because my far away is not actually far away, these angles are going to be terribly exaggerated.

02:01 But if you remember your derivations from class or your textbook, the point of going far away is that the waves from a and b become nearly parallel.

02:13 All right, and we're gonna pair that pink up with a green.

02:18 And so here is b reaching this point up here.

02:27 Oh my gosh, it's so nice, it actually gave me a straight line.

02:30 I don't know how it did that.

02:32 So what we have to consider here is that for destructive interference at this point, we know that b has to travel a half a wavelength further if they were in phase.

03:21 But because b is lagging by a quarter of a cycle, b needs to be able to travel an extra one quarter of a wavelength because of the lag.

03:37 So that's gonna be a path length difference that includes three quarters of a wavelength.

03:47 So i'm gonna call it the pld, the path length difference, is going to be for when b has to catch up with a, three quarters lambda.

04:11 And then we need additional integer values of lambda.

04:16 So three quarters lambda, the next one will be one and three quarters, two and three quarters, three and three quarters.

04:23 So we can write it as m plus three quarters quantity times lambda, where m will be zero, one, two, et cetera.

04:40 Now let's talk about what's going to go on with a, and then we'll dive into calculating our angles.

04:50 So what about below the center line? what is happening there? if we are considering below the center line, b will come out at this kind of direction.

05:08 And then we would have a being the one that's gonna travel, oh my gosh, i didn't quite get that angle correctly.

05:18 A is gonna be the one traveling further.

05:22 And for destructive interference at this point, well, a is already a quarter of a wavelength ahead.

05:41 So a needs only an extra quarter of a wavelength to be a full half a wavelength ahead of b to make destructive interference.

06:00 But we can't just do every quarter wavelength, because then we'd have a quarter, a half, three quarters, one wavelength.

06:12 We are gonna need a path length difference that is similar to what we wrote for b.

06:20 We're gonna need a quarter wavelength and then additional integer numbers of wavelengths.

06:25 So when a's path is a quarter of a wavelength longer, it will end up half a cycle ahead of b, which will create destructive interference.

06:35 And if you have one and a quarter, and two and a quarter, and three and a quarter, so we're gonna have m plus one quarter times lambda below the center line in our diagram.

06:50 And again, m is gonna be integers equal to zero, one, two, et cetera.

06:56 So now let's actually calculate these angles.

07:07 So recall that actually, before we get into that, we have one more thing we need to do, which is find out the wavelength of the sound waves anyway.

07:23 V equals lambda f, where lambda equals the velocity divided by the frequency...

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