Question

The human ear can detect continuous sounds in the frequency range from $20\mathrm{Hz}$ to $20,000\mathrm{Hz}$. Assuming that the speed of sound in air is $330{\mathrm{ms}}^{-1}$ for all frequencies, calculate the wavelengths corresponding to the given extreme frequencies of the audible range.

Answer 1

Step 1: Given data:

Speed of sound in air$=\mathrm{v}=330\mathrm{m}/\mathrm{s}$

Minimum frequency heard by the human ear $f_{\min }=20\mathrm{Hz}$

Maximum frequency heard by the human ear $f_{\max }=20\mathrm{kHz}=20000\mathrm{Hz}$

Step 2: Calculating the value of the minimum and maximum wavelength

As we know,

The wavelength of sound wave $=\lambda =\frac{\mathrm{v}}{f}$

Where, $f$ is the frequency of the sound wave and $v$ is the speed of sound in air.

As the wavelength is inversely proportional to the frequency, thus, for the maximum value of frequency, the minimum value of wavelength is obtained.

Since maximum frequency is heard by the human ear $f_{\max }=20\mathrm{kHz}=20000\mathrm{Hz}$.

So, the minimum wavelength of sound wave ${\lambda }_{min}=\frac{v}{f_{max}}$
$\therefore {\lambda }_{\min }=\frac{330}{20000}=16.5\times {10}^{-3}\mathrm{m}=16.5\mathrm{mm}$

Similarly, the minimum frequency is heard by the human ear $f_{\min }=20\mathrm{Hz}$.

The maximum wavelength of sound wave ${\lambda }_{\max }=\frac{\mathrm{V}}{f_{\min }}$
$\therefore {\lambda }_{\max }=\frac{330}{20}=16.5\mathrm{m}$
Hence, the wavelengths corresponding to the given extreme frequencies of the audible range will be $16.5mm$ and $16.5m$.