Problem

A certain pipe produces a fundamental frequency o…

02:01

Problem 27 Medium Difficulty

$\cdot$ The role of the mouth in sound. The production of sound during speech or singing is a complicated process. Let's concentrate on the mouth. A typical depth for the human mouth is
about $8.0 \mathrm{cm},$ although this number can vary. (Check it against your own mouth.) We can model the mouth as an organ pipe that is open at the back of the throat. What are the wavelengths and frequencies of the first four harmonics you can produce if your mouth is (a) open, (b) closed? Use $v=354 \mathrm{m} / \mathrm{s}$ .

Answer

a)
$\mathrm{I}^{\text { st }}$ harmonic $\lambda_{1}=2 L$
16 $\mathrm{cm}$,
$2^{\text { nd }}$ harmonic $\lambda_{2}=\frac{2 L}{2}$
8 $\mathrm{cm}$,
$3^{\text { rd }}$ harmonic $\lambda_{3}=\frac{2 L}{3}$
5.33 $\mathrm{cm}$,
$4^{\text { th }}$ harmonic $\lambda_{4}=\frac{2 L}{4}$
4 $\mathrm{cm}$,
$1^{\text { st }}$ harmonic frequency
2213 $\mathrm{Hz}$,
$2^{\text { nd }}$ harmonic frequency
4425 $\mathrm{Hz}$,
$3^{\text { rd }}$ harmonic frequency
6638 $\mathrm{Hz}$,
$4^{\text { th }}$ harmonic frequency
8850 $\mathrm{Hz}$

b)
$1^{\text { st }}$ harmonic wavelength
32 $\mathrm{cr}$
$2^{\text { nd }}$ harmonic wavelength
10.66 $\mathrm{cm}$
$3^{\mathrm{rd}}$ harmonic wavelength
6.4 $\mathrm{cm}$
$4^{\text { th }}$ harmonic wavelength
4.57 $\mathrm{cm}$
$1^{\text { st }}$ harmonic
11064$z$
$2^{\text { nd }}$ harmonic
3320 $\mathrm{Hz}$
$3^{\text { rd }}$ harmonic
5531 $\mathrm{Hz}$
$4^{\text { th }}$ harmonic
7746 $\mathrm{Hz}$

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Physics 101 Mechanics

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Video Transcript

We call that the production of sound during speeches. Singing is a complicated process, but let's concentrate on the mouth. Typical depth for the human mouth is about eight centimeters, although this very we come out of the mouth as an organ pipe that is open at the back of what are the wavelengths and frequency of the 1st 4 harmonics you can produce if your mouth is first opened and then closed, and we're gonna use speed of sound and air to be 354 meters per second? Because yes, we're assuming it's a little warmer in your mouth out in the So we have our mouth model as a an open, open organ pipe, given our length and speed of sound in air. So for an open, open pipe, the way the first wave length is too well and so you can plug everything in and we get 2213 hertz. The next harmonic is when the wavelength well, and so we get 4425 hertz. The next way way is when the way the next harmonicas, when the wavelength is that to well over three so we get 6638 Hertz and the final one. The final harmonic is when the wavelength is L over too. And so we get 8850 hertz. Now if we go on to look at the case where we have our mouth closed now we have an organ pipe with an open end at the back of our throat, and mouth is closed. So again we have the length and the speed of sound in air and for a closed for an open close pipe. The first harmonic is when the wait. Wait well and so that gives us 1106 hertz. Next. Harmonic is when the wavelength is that 4/3 that gives us 3319 hurts. Next, Harmonic. The wavelength is for well over five, and we get 5531 hurts. And for our fourth harmonic wave length is that four l over seven. So we get 7000 744 hurts

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