Question

The wavelength of a sound wave is reduced by $50\%$, then the percentage change in its frequency will be,
a) $100\%$
b) $200\%$
c) $400\%$
d) $800\%$

Answer 1

There is a basic relation between the wavelength and the frequency of a wave. This relation is valid for all waves. We will apply the changes mentioned, in the question, in the appropriate parameters and hence will obtain the answer.
Formula Used:
$v=\dfrac{\lambda }{T}$
$v=\lambda f$

Complete answer:
We know that the relation between the wavelength of the wave and its velocity is given by:
$v=\dfrac{\lambda }{T}$ ----(i)
Where, $v$ is the speed of the wave, $\lambda$ is the wavelength of the wave and $T$ is the time period of the wave.
We also know that the frequency of the wave is given by:
$f=\dfrac{1}{T}$ ----(ii)
Using equation (ii) in (i), we get:
$v=\lambda f$
$\Rightarrow f=\dfrac{v}{\lambda }$ ----(iii)
Now, consider $v$ is the speed of the sound wave, $\lambda$ is the wavelength of the sound wave and $T$ is the time period of the sound wave and $f$ is the frequency of the sound wave.
Using equation (iii), we can say that the frequency of original sound wave is given by:
$\Rightarrow f=\dfrac{v}{\lambda }$ ----(iv)
Now, when the wavelength of the sound wave is reduced by $50\%$, the new wavelength will be:
${{\lambda }_{new}}=\lambda -\dfrac{50}{100}\lambda$
${{\lambda }_{new}}=0.5\lambda$
There is no mention of the change in the velocity of the sound wave. Hence we are going to take it as a constant in both the cases.
So, the new frequency of the modified sound wave is given by:
${{f}_{new}}=\dfrac{v}{{{\lambda }_{new}}}$
$\Rightarrow {{f}_{new}}=\dfrac{v}{0.5\lambda }$
$\Rightarrow {{f}_{new}}=2\dfrac{v}{\lambda }$
This can also be written as:
${{f}_{new}}=2f$ -----(v)
Now, the percentage change in the frequency is given by:
$\dfrac{{{f}_{new}}-f}{f}\times 100$
$=\dfrac{2f-f}{f}\times 100$
$=100\%$

Hence, the correct option is option (a).

Note:
Any given equation may contain more than one variable. Whenever we need to compare any two variables in the equation, we tend to take the other variables as constant. If it is mentioned in the question, then well and good or else it is a general practice to take other variables as constant.