Two speakers, emitting identical sound waves of wavelength 2.0 m in phase with each other, and an observer are located as shown in

Fig. E35.5. (a) At the observer's location, what is the path difference for waves from the two speakers?

Video transcript

Everyone in this problem we have two speakers that are placed facing an audience. The speakers are playing a tone in phase With each other at a wavelength of 0.85 m. A member of the audience has the position shown in the Diagram and were asked to determine the path difference for the waves emitted by the speakers. Alright so we're talking about the path difference. Okay that's going to be the difference in the distance that the waves travel from one speaker to the audience member versus from the other speaker to the audience member. Okay? Alright so let's call this speaker on the left speaker L the speaker on the right speaker are and in order to find the path difference, we first need to find the distances from these speakers to our audience member. Alright so let's start with the left speaker, we're gonna call it D. L. And this is gonna be the distance from the left speaker two the audience number. Okay. All right now what is this distance? Well from the Diagram we can see that it's just 13 m. So d. L. is gonna be 13 m and we're gonna do the same for the right speaker. D. R. Is going to be equal to the distance from the right speaker to the audience member. Now, if you look at this right speaker, okay we can calculate this distance using a bit of triangle mouth. Yes we'll draw it out if we have the right speaker here. The left speaker here, we know the distance between the two of them is 10 m. Okay and then the distance between the left speaker and the audience member is 13 m. Okay? And the high pot noose of this triangle is going to be D. R. That distance. We're trying to find this is a right triangle, so we can use pythagorean theorem, we have that D. R squared. It's going to be equal to 10 m squared Plus m squared. This is going to be equal to 100 m squared plus 169 m squared, which gives us 269 m squared. We take the square root. When we take the square root, we're going to get the positive and the negative square root. In this case we're just concerned about the distance. So we're just gonna use the positive route. Ok? And we get 16.4 m. Okay? So now we have our distance from the left speaker to our audience member and the distance from our right speaker to the audience member. And the path difference is gonna be the absolute value of the distance between those two. Okay, How much more distance does the wave travel in one path than in the other? So path difference, we're gonna call it D. It's going to be the absolute value of D. R minus D. L. K. Or you could write D L minus D. R. Either way when you take the absolute value is going to give you the same answer. Okay? And because we've chosen d our first that's the bigger one, we can drop the absolute values because the answer is going to be positive, We get 16.4 m -13 m, which gives us a path difference of 3.4 m. Okay? And so the wave has to travel 3.4 m longer from the right speaker than the left speaker. So if we scroll down, we look at our answer choices. Okay? We see that we have answers. See the path difference is 3.4 m. That's it for this one. Thanks everyone for watching. See you in the next video.