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Singing in the shower! We all sound like great singers in the shower, due to standing waves. Assume that your shower is 2.45 (about 8 ft) tall and can be modeled as an organ pipe. (a) What will we have at the floor and ceiling, displacement nodes or antinodes? (b) What are the wavelength and frequencyof the fundamental harmonic for standing waves in this shower? (See the answer to Problem $28 .$ (c) What are the wavelength and frequency of the first two overtones for this shower?

a) nodesb) 4.9 $\mathrm{m}$, 70 $\mathrm{Hz}$c) $140 \mathrm{Hz}$ $2^{\text { nd }}$ overtone 210 $\mathrm{Hz}$

Physics 101 Mechanics

Chapter 12

Mechanical Waves and Sound

Periodic Motion

Mechanical Waves

Sound and Hearing

University of Michigan - Ann Arbor

University of Winnipeg

McMaster University

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So this is modeling singing in the shower. We're going to say that the length is going to be 2.45 meters. That would be the height of your shower, and then a is asking us here. The pipe is closed at both ends, but a pipe that is closed at both ends essentially models a pipe that is also open at both ends. It was the only thing that changes is when a pipe is closed at on ly one end. However, if it's closed at both ends or open at both ends, it doesn't matter. It acts. Standing waves will act the same in this pipe. So for party, there are displacement nodes at the floor and ceiling. Part B is asking us for the fundamental wavelength. This will simply be equal to two times out, or we can say 4.90 meters, and the fundamental frequency will be equal to the velocity, the speed of sound and air divided by Sig Landis of one. And this will be quick to 344 divided by 4.9 and this equal 70 point to hurts. Now we use the speed of sound and air. However, when you're taking a shower, it actually ah, a lot of the air. The air has a lot of water vapor in it because the air is of that. The water is evaporating at as you're taking a shower. So this is Ah, this is a rough approximation. We're using the velocity sound an heir. We should be using the velocity of sound in ah, moisture, heavy air or water vapor. However, we this approximation is good enough for the sake of this problem. And so the wavelength of the first overtone will be equal to the wavelength of the fundamental divided by two. So this will be equal to 2.45 meters and we can say the frequency of the first overtone will be equal to two times the frequency of the fundamental, and this will be 140 0.4 hertz. The second overtone or the third harmonic will be simply, ah, the fundamental wavelength divided by three. In this case, it will be 1.63 meters and then the frequency of the second overtone or the third harmonic will be three times the fundamental. So this will be 210.6 hertz. That is the end of the solution. Thank you for watching

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