Question

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 7.80cm , 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.
A) What are the wavelengths of the first four harmonics you can produce if your mouth is open?

?1,?2,?3,?4 = __ cm


B) What are the frequencies of the first four harmonics you can produce if your mouth is open? Use 350m/s.

f1,f2,f3,f4 = __ Hz


C) What are the wavelengths of the first four harmonics you can produce if your mouth is closed?

?1,?2,?3,?4 = __ cm


D) What are the frequencies of the first four harmonics you can produce if your mouth is closed? Use 350m/s.

f1,f2,f3,f4 = __ Hz

Answer 1

Answer:

a)  λ₁ = 15.6 cm,  λ₂ = 7.8 cm, λ₃ = 5.2 cm, λ₄ = 3.9 cm

b)   f₁ = 2243 Hz, f₂ = 4487 Hz, f₃ = 6730 Hz, f₄ = 8974 Hz

c)  λ₁ = 31.2 cm , λ₂ = 10.4 cm, λ₃= 6.24 cm, λ₄= 4.46 cm

d)   f₁ = 1121.8 Hz , f₂ = 3365 Hz ,  f₃ = 5609 Hz , f₄ = 7848 Hz

Explanation:

a) This simulation can be approximated as a tube that is open at both ends, in this case we have wave maximums at these points, the relationship between the wavelength and the length of the wave is

      λ = 2L

      λ₂ = (2L) / 2

      λ₃ = (2L) / 3

      λₙ = 2L / n

      n = 1, 2, 3 ...

Let's apply this equation to our case

      λ  = 2 7.80 cm

      λ₁ = 15.6 cm

      λ₂ = 2 7.8 / 2

      λ₂ = 7.8 cm

      λ₃ = 2 * 7.8 / 3

      λ₃ = 5.2 cm

      λ₄ = 2 7.8 / 4

      λ₄ = 3.9 cm

For the frequency we use the relationship

       v = λ  f

       f = v / λ  

       f1 = v / Lam

The wavelength in meters

      f₁ = 350 /15.6 10-2

      f₁ = 2243 Hz

      f₂ = 350 / 0.078

      f₂ = 4487 Hz

      f₃ = 350 / 0.052

      f₃ = 6730 Hz

      f₄ = 350 / 0.039

      f₄ = 8974 Hz

c) in this case the mouth is closed, therefore, at this point we have a node and the open part a belly (maximum)

      λ = 4L

      λ₂ = 4L / 3

      λ₃ = 4L / 5

      λₙ = 4L / n

      n = 1, 3, 5, 7, ...

We calculate

      λ₁ = 4 7.80 / 1

      λ₁ = 31.2 cm

      λ₂ = 4 7.8 / 3

      λ₂ = 10.4 cm

      λ₃ = 4 7.8 / 5

      λ₃= 6.24 cm

      λ₄ = 4 7.8 / 7

      λ₄= 4.46 cm

We calculate the frequencies the wavelength in meters

     f₁ = 350 / 0.312

     f₁ = 1121.8 Hz

     f₂ = 350 / 0.104

    f₂ = 3365 Hz

    f₃ = 350 /0.0624

    f₃ = 5609 Hz

    f₄ = 350 /.00446

    f₄ = 7848 Hz