Two speakers that are 15.0 m apart produce in-phase sound waves of frequency 250.0 Hz in a room where the speed of sound is 340.0 m>s. A woman starts out at the midpoint between the two speakers. The room's walls and ceiling are covered with absorbers to eliminate reflections, and she listens with only one ear for best precision. (a) What does she hear: constructive or destructive interference? Why?

Video transcript

Hey everyone in this problem, we have two speakers on a roadshow truck that are 11 m apart. A notification of frequency 850 hertz and in the same phases playing from the speakers, Okay, the speed of sound in the air at that instance is 346 m per second. The reflections from the walls of the truck are negligible. A programmer standing at the center of the two speakers, listening carefully while directly facing one of the speakers. The orientation means that both ears have the same distance measured from both speakers. Okay? And were asked whether the programmer detects constructive or destructive interference. Alright, so let's do a little diagram. Okay, we have our speaker on the left here, reverse speaker on the right here, And these are 11 m apart and then we have our person standing and listening. Okay? They're gonna be some distance from the left speaker. D. L. They're going to be some distance from the right speaker. D. R. Okay, and we're told that these distances are the same, so we're told that D. L. Is equal to D. R. Okay, now, when we're looking at our whether we have constructive or destructive interference, okay? We want to consider the path difference. Now the path difference, a delta X. Is going to be equal to the magnitude of the difference between the distance that our person is from the left speaker and the distances from the right speaker. Okay, well these two distances are equal. Okay, which means that our path difference is equal to zero. Alright, so our path difference is equal to zero? Which category does that fall under? Okay, well let's just write out what we have. Okay, if we have constructive interference, Okay. We want the path difference to be an integer multiple of the wavelength. Okay, so the path difference can be zero lambda two lambda dot dot dot. Okay. And we can write this in general as N lambda. Okay, for n equals 012 dot dot dot. Okay, if we have destructive interference, We want the path difference to be equal to a multiple of half the wavelength. So λ over two three lambda over two dot dot dot. Okay, we can write this as N lambda over two where we have N is equal to 135 dot dot dot. Okay. And what we'll see is that zero is included here? For delta X in the constructive case but not in the destructive case. Okay, We have this end value of zero included in constructive interference but not in destructive interference. And since we have a delta x of zero, that means that we're going to have constructive interference. Okay, Because that delta X is zero. So if we look at our answer choices here, we see that we have answer B Okay. The programmer is going to detect constructive interference because the path difference delta X is equal to zero. That's it for this one. Thanks everyone for watching. See you in the next video