Question

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.)

Answer 1

Answer:

$f_1=1011.76Hz\\f_2=2023.53Hz\\f_3=3035.29Hz$

Explanation:

Standing waves are not propagation waves but the different modes of vibration of a string, a membrane, etc. They are waves that result from the superposition of propagation waves that maintain a constant interference giving a new wave pattern. The frequency that a wave must have to give a stable standing wave is:

$f_n=\frac{nv}{2L}$

Where:

$L=Length\hspace{3}of\hspace{3}the\hspace{3}string \\v=Speed\hspace{3}of\hspace{3}propagation\hspace{3}of\hspace{3}the\hspace{3}wave\\n=Nth\hspace{3}harmonic$

Using this, we can estimate the first three standing-wave frequencies:

(Remember to convert cm to m)

$17cm*\frac{1m}{100cm} =0.17m$

$f_1=\frac{1*344}{2*0.17} \approx1011.76Hz$

$f_2=\frac{2*344}{2*0.17} \approx2023.53Hz$

$f_3=\frac{3*344}{2*0.17} \approx3035.29Hz$