Question

The human ear canal is, on average, 2.5cm long and aids in hearing by acting like a resonant cavity that is closed on one end and open on the other. The length of the ear canal is partially responsible for our sensitivities to certain frequencies. Use 340m/s for the speed of sound when performing the following calculations.
(a) What is the first resonant frequency?
(b) What is the wavelength at second resonance?
(c) How does the first resonant frequency change if the ear canal is full of water? (increases, shortens, no change or decreases?)

Answer 1

Answer:

(a) 3400 Hz

(b) 3.33 cm

(c) It increases

Explanation:

(a) The ear, being open at one end and closed at the other, is equivalent to a closed pipe. For a closed pipe, the first resonant frequency occurs when one quarter of the wavelength is equal to the length of the pipe.

$\dfrac{\lambda}{4}=l$

$\lambda = 4l$

Speed is given by

$v=f\lambda$ where f is the frequency.

$f=\dfrac{v}{\lambda}$

At the first resonant frequency,

$f=\dfrac{v}{4l}$

$f=\dfrac{340}{4\times2.5\times10^{-2}} = 3400\text{ Hz}$

(b) At second resonance,

$\dfrac{3}{4}\lambda=l$

$\lambda=\dfrac{4}{3}l$

$\lambda=\dfrac{4}{3}\times2.5\times10^{-2} \text{ m} = 3.33\times10^{-2} \text{ m} = 3.33\text{ cm}$

(c) It increases. Sound travels faster in a liquid than in gas. From the formula in (a), the resonant frequency increases with speed.