Q27E - University Physics with Modern Physics

Question

BIO The Human Voice. The human vocal tract is a pipe that extends about 17 cm from the lips to the vocal folds (also called “vocal cords”) near the middle of your throat. The vocal folds behave rather like the reed of a clarinet, and the vocal tract acts like a stopped pipe. Estimate the first three standing-wave frequencies of the vocal tract. Use v = 344 m/s. (The answers are only an estimate, since the position of lips and tongue affects the motion of air in the vocal tract.)

Step-by-Step Solution

Answer

The first three standing-wave frequencies of the vocal tract is  $505.88\text{\hspace{0.17em}Hz}$ ,   $1517.65\text{\hspace{0.17em}Hz}$ and  $2529.41\text{\hspace{0.17em}Hz}$ .

STEP-1: Given Data

Length of human vocal tract :  $L=17\text{\hspace{0.17em}cm}=0.17\text{\hspace{0.17em}m}$

Speed of sound in air:   $v=344\text{\hspace{0.17em}m}/\text{s}$

STEP-2: Standing Waves

When two waves moving in the opposite direction and having the same amplitude and frequency, are superimposed on each other, the resultant wave is known as a standing wave.

STEP-3: Calculating the first three frequencies of the vocal tract.

The first three standing waves in a stopped pipe are $\mathrm{n}=1,\mathrm{n}=3\text{\hspace{0.17em}and\hspace{0.17em}}\mathrm{n}=5$ . Here $n$  is the number of harmonics. The frequency of the $n^{th}$   harmonic is given as-

 $f=\frac{v}{\lambda }=n\frac{v}{4L}$

$\begin{array}{c}{f}_1=n\frac{v}{4L}=1\times \frac{344\text{\hspace{0.17em}m/s}}{4\times (17\times {10}^{-2}\text{\hspace{0.17em}m})}=505.88\text{\hspace{0.17em}Hz}\\ f_3=n\frac{v}{4L}=3\times \frac{344\text{\hspace{0.17em}m/s}}{4\times (17\times {10}^{-2}\text{\hspace{0.17em}m})}=1517.65\text{\hspace{0.17em}Hz}\\ f_5=n\frac{v}{4L}=5\times \frac{344\text{\hspace{0.17em}m/s}}{4\times (17\times {10}^{-2}\text{\hspace{0.17em}m})}=2529.41\text{\hspace{0.17em}Hz}\end{array}$ 

Therefore. the first three standing-wave frequencies of the vocal tract is $505.88\text{\hspace{0.17em}Hz}$ , $1517.65\text{\hspace{0.17em}Hz}$  and $2529.41\text{\hspace{0.17em}Hz}$ .