Question

A sound wave travels at a speed of $339\;{{\rm{m}} {\left/{\vphantom {{\rm{m}} {\rm{s}}}} \right.} {\rm{s}}}$. If its wavelength is $1.5\;{\rm{cm}}$, what is the frequency of the wave? Will it be audible?

Answer 1

Hint:To find the frequency, we will use the relation between the wavelength, frequency and velocity. We know that frequency is inversely proportional to the wavelength. In this question, wavelength and velocity are already given, we will substitute these in relation to find the frequency of the sound wave. Then to check whether it is audible or not, we will check for the audible range of the human ear.

Complete step by step answer:
Given:
The speed of sound is $v = 339\;{{\rm{m}} {\left/
{\vphantom {{\rm{m}} {\rm{s}}}} \right.
} {\rm{s}}}$.
The wavelength of the sound wave is $\lambda = 1.5\;{\rm{cm}}$.
We will assume frequency of sound wave as $\upsilon $
We will write the relation between the wavelength, frequency and velocity of sound as expressed:
$v = \upsilon \lambda $
We will rearrange the above expression as:
$\upsilon = \dfrac{v}{\lambda }$
In the above expression, we will substitute $339\;{{\rm{m}} {\left/
{\vphantom {{\rm{m}} {\rm{s}}}} \right.
} {\rm{s}}}$for $v$and $1.5\;{\rm{cm}}$for $\lambda $. This can b expressed as:
$\begin{array}{l}
\upsilon = \dfrac{{{\rm{339}}\;{{\rm{m}} {\left/
{\vphantom {{\rm{m}} {\rm{s}}}} \right.
} {\rm{s}}}}}{{1.5\;{\rm{cm}} \times \dfrac{{{{10}^{ - 2}}\;{\rm{m}}}}{{1\;{\rm{cm}}}}}}\\
\upsilon = 22600\;{{\rm{s}}^{{\rm{ - 1}}}}\\
\upsilon = 22600\;{\rm{Hz}}
\end{array}$
The frequency of the sound wave is $22600\;{\rm{Hz}}$.
To check whether this sound will be audible or not, we will check whether it comes in the range of audible range of human ears.
We know that the audible range for human ears is $20\;{\rm{Hz}}$ to $20,000\;{\rm{Hz}}$.
The frequency of sound waves in our solution is $22600\;{\rm{Hz}}$ and it is greater than the maximum frequency of the audible range of human ears. Hence, it will not be audible.

Note:We know that frequency and wavelength of the electromagnetic waves are inversely proportional. In this question, we are using this relation to find frequency. The velocity of the sound wave is a constant and it will be treated as constant of proportionality. We should have prior knowledge of the audible range of human ears to check whether it will be audible or not.