Two speakers that are 15.0 m apart produce in-phase sound waves o...

Question

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. (c) How far from the center must she walk before she first hears the sound maximally enhanced?

Accepted Answer

Hey everyone in this problem, we have two speakers in an open field that are separated by 20 m. A tune of 550 hertz from the same source is channeled to the speakers. The temperature is quite high, raising the speed of sound and air to 349 m/s. A recording microphone is located at the center of the two speakers and were asked what distance Okay, measured from the midpoint of the two speakers, will the microphone detect a first maximum gain of the sound? Alright, so let's just draw this, we have our speakers speaker one, We have speaker two and we know that they or 20 m now, we also have a microphone here. Okay, that is at the center of the two speakers. So the distance Between the speaker and the microphone is going to be 10 m On each side. It's halfway between the two. Alright, so that's what we have right now. Mhm. All right now, we're looking for a first maximum gain of sound. So we want to start with is to figure out is this constructive interference? Is this destructive interference? What situation are we dealing with? Okay, and then we can kind of look at what the next gain of sound is. So what what do you want? Okay, be the distance from Speaker one to the mic. Now, we're looking at these distances because we want to find the path difference. The path difference is going to help us figure out if we have constructive interference or destructive interference. Okay, so we need to find the distance from speaker one to the mic and Speaker two from the mic. Okay. We want to look at the difference in those distances to find our path difference. All right, So D. One the distance from speaker one to the mic. What we can see from our diagram that this is 10 m. Okay? And similarly, if we let D two B the distance from speaker to to the mic, this is also going to be 10 m to our path difference and we're gonna call it delta X again, is the difference in these distances and it's going to be the absolute value. Okay? We just care about the magnitude of the difference of the distance traveling from one Speaker to the mic to the other? Well, these distances are the same. So our path difference is just going to be zero m. Okay, If our path difference is zero Okay, this tells us that we have constructive interference. Okay, recall oops interference. Alright, recall that if we have a path difference of zero. Okay, that's included in the conditions for constructive interference. It is not included in the construct in the um conditions for destructive interference. Okay, So with constructive interference, our path difference okay, is going to be zero lambda to lambda dot dot dot. Okay. And we can just write this as n lambda. Okay, For n equals 012. Okay. And so we're multiple of lambda and integer multiple of lambda. For our path difference and we see that zero is included there. So we're dealing with constructive interference. All right. No, the distance we've written here, it's the distance from the speaker to the mic. Okay. We want to figure out what is the distance measured from the midpoint of the two speakers. Okay. We want to consider if this mike is moving okay if we end up with a different path difference than zero? Right now, we have a path difference of zero. We want to find the first maximum gain of sound. So if we have a different path difference, Okay, So let's let D M. B. The distance from the well then D one, the distance from speaker one to the microphone is going to be 10 m Okay? Plus the distance from the midpoint and we're going to consider this mike to be going to the right kind of positive. So if we move this mic a little bit to the right Speaker one is going to be D one. 10 m plus D. M from that microphone. Speaker two. Okay. Will therefore be 10 m minus DM Okay, if we move the speaker to the right speaker one is going to be further away from it. Speaker two is going to be closer to it. Okay, So we're going to reduce the distance from of D two and increase the distance of D one. Okay. And that's taking DM positive to be to the right. If you were to switch the sign and say that to the left is positive, it would be the opposite case. Okay. All right, so we have our distance one. We have our distance to in general if we're moving this mic around in the middle. Okay, what's the path difference gonna be? Well, the pack difference, delta X. Is still Do you want minus D to the absolute value? Okay. That means we're gonna have 10 m plus dm minus 10 m minus deal. Okay. The 10 m minus 10 m that goes away, that's gonna be zero m and we have D m minus negative diem. So that's going to give us two terms of distance D. M. Okay, So two times the distance from the midpoint is our path difference. Now we know that we want our path difference delta X to be equal to n. Lambda. And we also know that our path difference delta X. Is equal to two D. M. This tells us that we want two times the distance from the midpoint to be equal to some integer multiple of lambda. They were in a 012 dot dot dot. This tells us that the distance from the midpoint that we're looking for should be And over two times λ. Now we know that N equals zero is our current situation. Okay? That's when our path difference is zero, the distance from the midpoint is zero. Okay, so that's our current situation. So the next one where we're going to get the first maximum gain of sound is going to be N equals one. So the first maximum gain of seven is gonna be one. N is equal to one. In this case when N is one, the distance from the midpoint becomes one half. It's a wavelength clip. Alright well we're not given the wavelength lambda. Let's recall that we can relate the wavelength lambda to the frequency into the speed. Okay so we get one half lambda which is going to be the the speed divided by f the frequency were given both the speed and the frequency and the problem. Okay one half were told that the speed of sound is 349 m per second. Okay, the air is hot, changes that speed a little bit And we have a frequency of 550 hertz. Again given in the problem. Well well we work this out, we get that this is equal to 0. 73 m. Okay so the distance from the midpoint for the first maximum gain of sound is going to be 0.3173 m. If we look at our answer choices. Okay, we see that these are all in cm. So let's just go back, convert this 2cm so that we can compare it to our answer choices. Okay, so we get 0.3173 m. And we're gonna multiply by 100 centimeters per meter. Okay. The unit of meter divides out and we are left with 31.73 centimeters. Okay. To go from meters to centimeters. We multiplied by 100 and we get Our distance from the midpoint of 31.73 cm. Alright, so back up to answer traces and we can see that we have answer choice. E. Okay, we found the distance measured from the midpoint of the two speakers where the microphone will detect a first maximum gate of sound is E. 31.7 centimeters. Thanks everyone for watching. I hope this video helped see you in the next one.

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