Question

A person has a hearing range of $20Hz$ to $20kHz$. What are the typical wavelengths of sound waves in air corresponding to these two frequencies? Take the speed of sound in air as $344m{s}^{-1}$.

Answer 1

Step 1: Given data.

Least frequency audible by a person, $f_1=20Hz$

Maximum frequency audible by a person, $f_2=20kHz$

Speed of sound in air, $v=344m{s}^{-1}$

Wavelength corresponding to $f_1$ is ${\lambda }_1=?$

Wavelength corresponding to $f_2$ is ${\lambda }_2=?$

Step 2: Calculate the wavelength corresponding to the minimum audible frequency.

The wavelength corresponding to $f_1=20Hz$ is calculated below:

$\begin{array}{rcl}{\lambda }_1& =& \frac{v}{f_1}\\ & \Rightarrow & {\lambda }_1=\frac{344m{s}^{-1}}{20Hz}\\ & \Rightarrow & {\lambda }_1=17.2m\end{array}$

Step 3: Calculate the wavelength corresponding to the maximum audible frequency.

The wavelength corresponding to $f_2=20kHz$ is calculated below:

$\begin{array}{rcl}{\lambda }_2& =& \frac{v}{f_2}\\ & \Rightarrow & {\lambda }_2=\frac{344m{s}^{-1}}{20kHz}\\ & \Rightarrow & {\lambda }_2=\frac{344m{s}^{-1}}{20\times 1000Hz}\\ & \Rightarrow & {\lambda }_2=0.0172m\end{array}$

Hence, the typical wavelengths of sound waves in air corresponding to $20Hz$ is $17.2m$ and corresponding to $20kHz$ is $0.0172m$.