Question

A person has a hearing range of $20Hz$ to $20kHz$ . Find the typical wavelengths of sound waves corresponding to these two frequencies. If the speed of sound is $344m{{s}^{-1}}.$

Answer 1

Hint: We will use the relationship of wavelength of sound in terms of velocity of the sound wave and frequency of the sound wave. We also know that the normal range of hearing is in between $20Hz$ to $20kHz$ . This range is also called audible range.

Formula Used:
In order to find the correct solution we will apply the following formula:-
$\lambda =\dfrac{v}{\nu }$ .

Complete answer:
From the question above we have the following parameters with us:-
Speed of sound wave, $v=344m{{s}^{-1}}$ .
First value of frequency, ${{\nu }_{1}}=20Hz$ .
Second value of frequency, ${{\nu }_{2}}=20kHz=2000Hz$ .
Now, wavelength at first frequency is represented as ${{\lambda }_{1}}$ and given as follows:-
${{\lambda }_{1}}=\dfrac{v}{{{\nu }_{1}}}$ …………. $(i)$
Putting the respective values in the equation $(i)$ we get
${{\lambda }_{1}}=\dfrac{344}{20}$
$\Rightarrow {{\lambda }_{1}}=17.2m$ …………. $(ii)$
Now, wavelength at the second frequency is represented as $\lambda {}_{2}$ and given as follows:-
${{\lambda }_{2}}=\dfrac{344}{2000}$
$\Rightarrow {{\lambda }_{2}}=1.72\times {{10}^{-2}}m$ .
Therefore, the typical wavelengths are \[{{\lambda }_{1}}=17.2m\] and ${{\lambda }_{2}}=1.72\times {{10}^{-2}}m$ at the frequencies $20Hz$ and $20kHz$ respectively.

Additional Information:
We should know the definitions of wavelength, frequency and speed of sound waves first to get the better grip on the topic.
The distance between the adjacent similar parts of the sound wave is called wavelength.
The number of oscillations per unit time is called frequency of the sound wave.
The speed at which sound wave travel is called speed of sound or the distance travelled by sound waves per unit time is called sound speed.

Note:
It should be noted that speed of sound in a medium does change with the change in temperature also. Formula should be used correctly without any confusion as the symbols of velocity and frequency are similar to each other. It should be noted that sound is a longitudinal wave.