Question

A source of sound producing wavelength $50\,{\text{cm}}$ is moving away from a stationary observer with ${\dfrac{1}{5}^{th}}$ speed of sound. Then what is the wavelength of sound heard by the observer?
A. $55\,{\text{cm}}$
B. $40\,{\text{cm}}$
C. $60\,{\text{cm}}$
D. $70\,{\text{cm}}$

Answer 1

Use the formula for frequency of the sound wave heard by the stationary observer when the source of sound is moving away from the stationary observer. Determine the frequency of the sound wave heard by the observer. Also use the formula for velocity of a wave in terms of frequency and wavelength of the wave. Using this formula, determine the wavelength of the sound wave heard by the observer.

Formulae used:
The formula for frequency $f'$ of the sound heard by a stationary observer when the source of sound is moving away from the observer is
$f' = \dfrac{v}{{v + {v_S}}}f$ …… (1)
Here, $f$ is the frequency of the source, $v$ is velocity of the sound wave and ${v_S}$ is velocity of the source.
The velocity $v$ of a wave is given by
$v = f\lambda$ …… (2)
Here, $f$ is frequency of the wave and $\lambda$ is wavelength of the wave.

Complete step by step answer:
We have given that the wavelength of the wave produced by the source is $50\,{\text{cm}}$.
$\Rightarrow\lambda = 50\,{\text{cm}}$
Also we have given that the velocity of the source is of the velocity $v$ of sound.
${v_S} = \dfrac{1}{5}v$
Substitute $\dfrac{1}{5}v$ for ${v_S}$ in equation (1).
$f' = \dfrac{v}{{v + \dfrac{1}{5}v}}f$
$\Rightarrow f' = \dfrac{v}{{\dfrac{6}{5}v}}f$
$\Rightarrow f' = \dfrac{5}{6}f$

Rearrange equation (2) for wavelength.
$\lambda = \dfrac{v}{f}$
Rewrite the above equation for the wavelength $\lambda '$ of the sound wave heard by the observer.
$\lambda ' = \dfrac{v}{{f'}}$
Substitute $\dfrac{5}{6}f$ for $f'$ in the above equation.
$\lambda ' = \dfrac{v}{{\dfrac{5}{6}f}}$
$\Rightarrow \lambda ' = \dfrac{6}{5}\dfrac{v}{f}$
Substitute $\lambda$ for $\dfrac{v}{f}$ in the above equation.
$\Rightarrow \lambda ' = \dfrac{6}{5}\lambda$

Substitute $50\,{\text{cm}}$ for $\lambda$ in the above equation.
$\Rightarrow \lambda ' = \dfrac{6}{5}\left( {50\,{\text{cm}}} \right)$
$\Rightarrow \lambda ' = 60\,{\text{cm}}$
Therefore, the wavelength of the wave heard by the stationary observer is $60\,{\text{cm}}$.

Hence, the correct option is C.

Note: The students should keep in mind that if the source of sound will be approaching the stationary observer, there must be addition of velocity of sound and velocity of the source in formula for frequency heard by the stationary observer. In the present question, the source is moving away from the stationary observer. Hence, there is a negative sign in the denominator of the formula of frequency heard by the observer.