Standing sound waves are produced in a pipe that is 1.20 m lon...

Question

Standing sound waves are produced in a pipe that is 1.20 m long. For the fundamental and first two overtones, determine the locations along the pipe (measured from the left end) of the displacement nodes and the pressure nodes if the pipe is closed at the left end and open at the right end.

Accepted Answer

Hey everyone in this problem. A plumber whistles into the right open end of a metal tube of length three m and generates standing waves were asked to draw a diagram showing the locations of the displacement nodes and anti notes along the tube for the fundamental and first overtone. Okay. We're told that the left end of the tube is closed. All right. So the plumbers whistling into an open end to the other end of the tube is closed. Okay, So this is going to be like a stopped pipe situation. Okay? We have one end open and one end closed. We think of, okay, we have a stopped pipe. Now, what's going to happen at the open end? Okay. We know that when we have an open end and we're gonna have a displacement anti node and also recall that at the closed end we're gonna have the opposite. We're gonna have a displacement. Okay, note. Alright, so we know what this looks like at the two ends at the closed end. We're gonna have a displacement node at the open end. We're gonna have a displacement anti note. Let's go ahead and figure out what happens between those two ends. All right. So when we have a stopped pipe, okay, We'll call it the possible wavelengths will be given by lambda and 24 L over N where L is the length of the pipe. Okay. And this is for odd values of it. Okay, so we have n equals 135 here, we get odd harmonics. No even harmonics. So the first one we're asked to look into is a fundamental mode. Okay, so in the fundamental mode Here we have an is equal to one. This means that lambda one is going to be equal to four times L one is just divide by one. and the length of our pipe we're told is three m or the length of the Metal tube. So three m We have four times 3 m which tells us But our wavelength is going to be 12 m. Alright, so now we have to figure out what's going on throughout this pipe. Well we know that X equals 0m. Okay, we're gonna have a note, we're gonna say that X equals zero m is the closed end and we have a displacement note. Alright, now the next notable place. Okay, once we go a quarter of the wavelength, if we started a node, okay, let's imagine a sine wave kind of like this and you can imagine to the two waves going like this. Woo. Okay. Not a great drawing but help show the point. So we're at a displacement node. Okay, we know that the next displacement note is gonna be at half of the wavelength and then we have another displacement node at the full wavelength for the anti notes. They're going to come at the quarters if we're at a displacement No, then at a quarter of the wavelength we get a displacement anti node and then again at three quarters of the wavelength. Okay, you can think of when you if you can't remember or recall where you should be looking. Just think of if you have a sine wave or cosine wave or wave that you know the information about and think about where you would get those nodes and anti notes. So at a quarter of the wavelength, Okay, so at 1/4 of the wavelength we found which was m, this is going to be at three m. We know we have now a displacement anti note. And what you'll notice is that this three m, that's the end of our price near the end of our metal tube. So we have a displacement node and then we have a displacement anti node. And that's it throughout that length of the pipe. So if we draw this out With the closed end here, this is at x equals zero m. And then we have the open end of the pipe down here that X equals three m. The closed end is a displacement node and then the open end is a displacement anti note. And it's gonna look something like this. Alright, so that's for our fundamental mode. Now we were also asked about the first over to now for the first overtone and we're gonna have N is equal to three. And be careful here. It can be really easy to say N is equal to two. But remember when we're looking at wavelengths for a stopped pipe, we can only take end values that are odd. Okay, so after the first harmonic or the fundamental mode you're into N equals three for your first overtone. So we have N. Is equal to three, which tells us that lambda it's gonna be equal to four L. Over three Is equal to 4/ the length of the pipe, which is three m. And so this is gonna be equal to four m. Alright, so doing the same thing again. Okay, At X equals zero m. We know we have a displacement note. Okay, because that's the closed end of our pipe. Now we're going to look at a quarter of the wavelength at a quarter of the wavelength. So a quarter of 4m which is going to be equal to one m. We have a displacement anti note. Alright, now let's look at half. Okay, we're only at one m. Our pipe is three m long. So what's happening past one m? Well, at half of our wavelength, so a half times four m, Which is at two m, we're going to have another displacement note. Okay, think of again, think of the sign curve halfway through world. Another note. Alright, we're still at two m or pipe is three m long. So let's go ahead and figure out what else is going on At three quarters of our wavelength. So at three quarters times are four m. This is going to be three m and we have a displacement anti node. And this makes sense because we said that the open end of our pipe should have a displacement anti node. And we found that at three m which is the end of our pipe, we have a displacement anti note. So that works out And now we can go ahead and draw. So again x equals zero m at this end, X equals three m at this end and then we're just going to break it up. So we have one m 2 m. Now we know that we have a note at X equals zero m displacement node. We know that we also have a displacement note at X equals two m. And then we know that we have anti nodes at one m and three m. So we're going to be kind of here here here here here. So if we draw this we're gonna get something that looks like this. Alright. So we have our first overtone, we did our fundamental mode. Now we can go back up to the top and look at the answer choices and see which ones match what we found. Alright, so we can see that we found our fundamental mode, we have a note at the closed end, an anti note at the open end and nothing else happens in between. So we're looking at either option A or option B. And then for the first overtone we found that we had two nodes and two anti nodes within the length of that metal tube. Okay, so we are looking at options A. These match the diagrams that we drew below. Thanks everyone for watching. I hope this video helped see you in the next one.

Standing sound waves are produced in a pipe that is 1.20 m long. For the fundamental and first two overtones, determine the locations along the pipe (measured from the left end) of the displacement nodes and the pressure nodes if the pipe is closed at the left end and open at the right end.

Key Concepts

Standing Waves

Boundary Conditions in Pipes

Harmonics and Overtones

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