Question

A person has a hearing range from 20 Hz to 20 kHz. What are the typical wavelengths of sound waves in air corresponding to these two frequencies? Take the speed of sound in air as 344 m/s.

Answer 1

Hint: Velocity of the sound in air is equal to the product of frequency of the sound and the wavelength of the sound. Using this relation and given values, we can find the wavelength range of the human ear.

Formula used:
The wavelength of a sound wave is given as
$\lambda ={\displaystyle \frac{\text{v}}{\nu }}...\text{(i)}$
where $\lambda$ is the wavelength of the sound wave, $\nu$ represents the frequency of the sound wave while v stands for the velocity of sound waves.

Detailed step by step solution:
Frequency of a wave is defined as the number of times the wave oscillates about its mean position per second. It is equal to the reciprocal of the time period of the wave.
Wavelength of a wave is defined as the distance covered by a wave in one time period. It can also be defined as the distance between two crests or two troughs of a wave.
Human ears are sensitive only to a certain range of frequencies of sound waves.
We are given the hearing range of a person which is from 20 Hz to 20 kHz. Therefore, we can write
$$\begin{gathered} {\nu }_1=20Hz \\ {\nu }_2=20kHz=20\times 1000Hz=2\times {10}^{4}Hz \end{gathered}$$
We are given the velocity of sound waves in air to be 344 m/s.
$\therefore \text{v}=344m/s$
Now we can calculate the wavelength of the given frequencies using the equation (i) in the following way.
$$\begin{gathered} {\lambda }_1={\displaystyle \frac{\text{v}}{{\nu }_1}}={\displaystyle \frac{344}{20}}=17.2m \\ {\lambda }_2={\displaystyle \frac{\text{v}}{{\nu }_2}}={\displaystyle \frac{344}{2\times {10}^4}}=0.0172m=1.72cm \end{gathered}$$
Hence the hearing range of a person in terms of wavelength is from 1.72cm to 17.2m.

Note: The sound waves are longitudinal waves and are characterized by the constriction and rarefaction in the medium in which the sound waves travel. Sound waves cannot travel without a medium. The range of sound waves for humans signifies that eardrum cannot vibrate at frequencies below or above the given range.