The three identical loudspeakers in FIGURE P17.71 play a 170 Hz tone in a room where the speed of sound is 340 m/s . You are standing 4.0 m in front of the middle speaker. At this point, the amplitude of the wave from each speaker is a.

c. When the amplitude is maximum, by what factor is the sound intensity greater than the sound intensity from a single speaker?

Video transcript

Hey, everyone in this trouble, we have three students doing, doing a piano show and they're sitting as portrayed in the face. So we have the first student on the left, they're all in a row and each student is four m away from the other. And we have the principal that's sitting three m in front of that middle student. We're told that the amplitude of the sound they play is B at a frequency of 170 Hertz. And that's for each student. And we're asked to determine the ratio of the maximum intensity of sound to the sound intensity of a single piano. We're told that the sound waves are emitted in face. We can consider the speed of sound to be 340 m per second. And all of the measurements are done at the position of the principle. We have four answer choices. Option A one third, option B 1/9 option C nine, divided by one and option D three divided by one. We're asked to find this ratio of intensities. Let's recall that the intensity of a sound wave K is proportional to the square of the amplitude. So we can write that I is equal to C multiplied by a square where A is the amplitude A and C is just some proportionality constant, right? So if we think about the single piano, OK, that's what we're told most information about. If we think about the single piano, the intensity of a single piano I single, gonna be equal to that constant. C multiplied by the amplitude, which we're told is B OK? So we get C B squared and we're not gonna worry about this proportionality constant just yet because we're doing a ratio, that proportionality constant is actually going to eventually divide it. So you don't need to worry about it and you'll see that step for more later. OK? All right. So we have our intensity for a single piano. We wanna compare that to the maximum intensity of sound. In order to find the maximum intensity, we need to find the maximum amplitude. Mhm. All right. Now, the max amplitude, we're gonna have these three pianos playing and we need to determine what happens at the principle. OK. If we have perfect constructive interference, we know that the amplitude will be three multiplied by the amplitude of each piano because we have three pianos. OK? That way those waves are gonna be arriving in phase at the principle. We're told that they're omitted in phase. But what happens at the principal? OK. So let's look at constructive interference and we're gonna start by looking at constructive interference between student one and student two. And we're gonna label them from left to right. So we have student one on the left, student two in the middle student three on the right. OK. So let's look at constructive interference, right? And this is gonna be between student one and two and it will occur if the path difference delta R, which we can write as the absolute value D one minus D two or D one is the distance from piano one to the principle. And D two is the distance from piano two to the principle is equal to N lambda where N is an integer, 012 and onwards. OK. So if it's an integer multiple of the path difference, right? Well, what is the path difference in this case? We know that D two, the distance between piano two, which is that middle piano and the principle is just three m. What about D one? We can imagine drawing a triangle this now the one side length is three m that distance from piano two to the principle, the top side length is four m. The distance between the two pianos and D one is gonna be the hypotenuse. So we can write that D one squared is gonna be equal to D two squared, a distance between the piano and the principal plus DB squared, which is gonna be the distance between pianos. All right. So substituting in our values, we want to find D one, we get that D one squared is equal to three squared plus four squared. And, and you'll recognize this as a perfect square, perfect triangle. We have D one squared is equal to 25. And so the distance D one is going to be equal to five. And again, this is in meters. So our path difference delta R is there gonna be the absolute value of five m minus two m. We want this to be equal to N multiplied by the wavelength. So we aren't given information about the wavelength, but we do know the speed and the frequency and recall, we can write the wavelength lambda as the speed divided by the frequency. And let's actually do that above first. So we're gonna do that in red and we know that the wavelength is gonna be equal to the speed divided by the frequency. We're told that the speed of sound here is 340 m per second. The frequency is 170 Hertz. And so the wavelength is gonna just be two m. So we can substitute that into our equation. We substituted D one already, we're gonna substitute our wavelength of two m. And on the left hand side, I have made a mistake here, we have five m minus two m. This should be five m minus three m. OK. That distance from the principle to piano two is three m. So on the left hand side simplify and we just get two m on the right, we have N multiplied by two m. OK? And this is true if N is equal to one. OK. So this works, we're gonna give that a check mark. We have constructive interference between student one and two, perfect constructive interference with an end value of one. Now, we want to do the same between the next two students. So what we're gonna do is we're gonna do the same thing, but we're gonna look at students one and three. OK? So between student one and three and again, it's going to occur if the path difference delta R which in this case is gonna be the absolute value of D one minus D three is equal to N lambda or N is an in. Now let's look at our diagram and think about this path difference, right? We're talking about D one and D three and the distance between piano three and the principle gonna be this distance here. And you can see that that's exactly the same as the distance do you want? OK. They're both, we both have to go three m up and four m to the side to get to those pianos. The difference of those distances is going to be zero. OK? Those distances are exactly the same. So when we subtract them, we get zero. So we have that zero m is going to be equal to N multiplied by that wavelength we found before which is two m and that zero m comes from our path difference, our two m comes from our wavelength. And so this is gonna be true if N is equal to zero. OK. So again, this is a true statement, we have constructive interference. So if we have constructive interference between student one and two and between student one and three, that means that we have interference, that is perfectly constructive between all three of those students. OK. So we have perfectly constructive interference. And what that tells us is that the maximum amplitude A max, it's just gonna be equal to three multiplied by the amplitude of each PM, right? Because we have three PM and we know that the amplitude of a single panel is equal to B and so our maximum amplitude A math is gonna be equal to three B, all right. So let's remember what we were doing. We're looking at a ratio of intensities in order to find the intensity, the maximum intensity we needed to find the maximum amplitude. So we looked at this constructive interference to make sure we had perfect constructive interference that told us about our maximum amplitude. Now we can get back to our maximum intensity. So the maximum intensity I maxed it's gonna be equal to C multiplied by the maximum amplitude A max squared. So it's gonna be equal to C multiplied by three B squared, which is equal to nine C B squared. Now the question is asking for the ratio of maximum intensity to the intensity of a single pian. So what we're looking for is I max divided by I single. OK. And always double check which order the ratio is asking you to calculate it. And. Ok. Because if you calculate I single divided by I max, you're gonna get the reciprocal. OK. So the answer could change. Now our maximum intensity nine CV squared divided by the intensity of a single piano which we found earlier to be C B squared. OK. And here is that point where those proportionality constant CS are gonna divide it. OK? We have C in the numerator, C in the denominator, they go away. We don't need to worry about them. Now, our B squared term is also gonna divide out and we're gonna be left with just nine divided by one. So that is the ratio we were looking for the ratio of the maximum intensity of sound to the sound intensity of a single piano. And if we compare this to our answer choices, we can see that this corresponds with answer choice. C Thanks everyone for watching. I hope this video helped you in the next one.