Question

A light wave and a sound wave have same frequency $f$ and their wavelengths are respectively ${\lambda _L}$ and ${\lambda _S}$ then
A. ${\lambda _L} = {\lambda _S}$
B. ${\lambda _L} > {\lambda _S}$
C. ${\lambda _L} < {\lambda _S}$
D. ${\lambda _L} = 2{\lambda _S}$

Answer 1

Use the formula for velocity of the wave. This formula gives the relation between the velocity of the wave, frequency of the wave and wavelength of the wave. Rewrite this formula for the wavelengths of the light and sound wave and take their ratios. Using the concept that velocity of light wave is very greater than the velocity of sound wave, determine the relation between the wavelengths of the light and sound wave.

Formula used:
The velocity $v$ of a wave is given by
$v = f\lambda$ …… (1)
Here, $f$ is frequency of the wave and $\lambda$ is wavelength of the wave.

Complete step by step answer:
We have given that the frequency of a sound wave is same as that of the light wave which is $f$.We have also given that the wavelength of the light wave is ${\lambda _L}$ and the wavelength of the sound wave is ${\lambda _S}$. We have asked the relation between the wavelengths of the light wave and sound wave.

Let $c$ and $v$ be the speeds of the light wave and sound wave.Let us first rewrite equation (1) for wavelength of the wave.
$\lambda = \dfrac{v}{f}$ …… (2)
We can write the formula for speed of the light wave using equation (1).
${\lambda _L} = \dfrac{c}{f}$ …… (3)
We can write the formula for speed of the sound wave using equation (1).
${\lambda _S} = \dfrac{v}{f}$ …… (4)

Let us now determine the relation between the wavelengths of the light wave and sound wave.Divide equation (3) by equation (4).
$\dfrac{{{\lambda _L}}}{{{\lambda _S}}} = \dfrac{{\dfrac{c}{f}}}{{\dfrac{v}{f}}}$
$\Rightarrow \dfrac{{{\lambda _L}}}{{{\lambda _S}}} = \dfrac{c}{v}$
We know that the speed of light waves is very greater than the speed of sound waves.
$c > > > v$
$\Rightarrow\dfrac{c}{v} > > > 1$
Hence, we can write equation for the wavelengths of the light wave and sound wave as
$\therefore {\lambda _L} > {\lambda _S}$
The above expression gives the relation between the wavelength of the light wave and sound wave.

Hence, the correct option is B.

Note: The students should use the formula for the velocity of the wave correctly. If this formula is not used correctly the final relation between the wavelengths of the light wave and sound wave will be incorrect. Also the students should correctly use the relation between velocities of the light and sound wave which helps to determine the relation between the wavelengths of the waves.