Question

If the frequency of sound waves in air is doubled. Its wavelength.
A. Is doubled
B. Increases 4 times
C. Decreases 4 times
D. Is halved

Answer 1

The velocity of a wave can be defined in terms of the frequency and the wavelength of the wave. Again, the velocity of a wave in a medium is always constant. From the relation of the velocity, frequency and the wavelength of the wave, we can find the required answer.

Complete step by step answer:
The speed of the sound wave in a medium is always constant but in different medium the velocity will be different. The speed of the sound wave can be mathematically related to the frequency and the wavelength of the sound wave in the medium.
We can write the speed of sound waves as the product of the frequency and the wavelength of the sound wave.
Let, the velocity of sound waves in air is v. Since, the velocity of sound waves in a medium is always constant, the velocity of sound waves in air will also be constant irrespective of the frequency and the wavelength.

Let, the frequency of a sound wave in air is$\nu$ and the wavelength of the sound wave is $\lambda$ .
So, we can write the velocity of the sound wave as,

$v=\nu \lambda$
So, $\lambda =\dfrac{v}{\nu }$
Now, when the frequency of the sound wave is doubled, let the wavelength of the sound wave will be ${{\lambda }_{f}}$

We can write,

$v=2\nu {{\lambda }_{f}}$
So, ${{\lambda }_{f}}=\dfrac{v}{2\nu }$

So, taking the ratio of the two wavelengths we get,

$\begin{aligned} & \dfrac{{{\lambda }_{f}}}{\lambda }=\dfrac{1}{2} \\\ & {{\lambda }_{f}}=\dfrac{\lambda }{2} \\\ \end{aligned}$

So, the wavelength of the sound wave in air will be halved when the frequency is doubled.
Hence, the correct answer is option D.

Note:
A sound wave can be called a pressure wave. It is a longitudinal wave. The velocity of the sound wave is different in different mediums depending on the density of the medium. But in a specific medium, the velocity of the wave is always constant.