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Chapter 2: Q28E (page 539)

The Vocal Tract. Many opera singers (and some pop singers) have a range of about21/2 octaves or even greater. Suppose a soprano鈥檚 range extends from A below middle C (frequency 220 Hz) up to E-flat above high C (frequency 1244 Hz). Although the vocal tract is quite complicated, we can model it as a resonating air column, like an organ pipe, that is open at the top and closed at the bottom. The column extends from the mouth down to the diaphragm in the chest cavity, and we can also assume that the lowest note is the fundamental. How long is this column of air if v = 354 m/s? Does your result seem reasonable, on the basis of observations of your own body?

Short Answer

Expert verified

The length of the air column in the vocal tract is 0.39 m.

Step by step solution

01

STEP 1 Concept of the frequency of standing wave in a stopped pipe

A pipe which has a closed end is given by

f=nv4L

where,f is the frequency of nth harmonic, v is the velocity of the wave, nrepresents the number ofharmonic (n 鈥� 1, 3, 5, ...), L is the length of the pipe.

02

Calculate the length of the column

The frequency of thenth harmonic is given by the formula-

f=nv4LL=nv4f=1脳344m/s4脳220Hz=0.39m

Therefore, the length of the column is 0.39 m.

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