Question

The relation between frequency (n) and wavelength $\left(\lambda \right)$ is given by ( v is velocity, n is frequency and T is the time-period)
$$\begin{gathered} (a)\text{ v = n}\lambda \\ \text{(b) n = }{\displaystyle \frac{\lambda }{v}} \\ (c)\text{ v = }{\displaystyle \frac{n}{{\lambda }^2}} \\ (d)\text{ n = }{\displaystyle \frac{T}{\lambda }} \end{gathered}$$

Answer 1

Hint: In this question use the relationship between the velocity of any wave travelling in the medium, wavelength and frequency that is $v=n\lambda$, moreover frequency is also defined as the inverse of time period that is $n={\displaystyle \frac{1}{T}}$. This will help establish the required relationship.

Complete step-by-step answer:

As we know that frequency (n) is the inverse of the time period (T).

$\Rightarrow n={\displaystyle \frac{1}{T}}$ Hz........................ (1)

Now as we know that the velocity of any wave travel in the medium is the product of the wavelength of the wave and the frequency of the wave.

$\Rightarrow v=n\lambda$ ........................... (2)

Where, v = velocity of the wave, n = frequency of the wave and $\lambda$ = wavelength of the wave.
Frequency is often measured in Hertz (Hz) or in per second (/sec), wavelength is often measured in meter (m).
So the velocity is measured in (m/s).

Now from equation (1) and (2) we have,

$\Rightarrow v={\displaystyle \frac{\lambda }{T}}$ m/s.

So this is the relation of velocity in terms of wavelength ($\lambda$) and time period (T).
So velocity of the wave is directly proportional to the wavelength of the wave and inversely proportional to the time period of the wave.
So in general velocity is directly proportional to the wavelength, so for a constant frequency if the wavelength of the wave is doubled so the velocity of the wave is also doubled.
So this is the required answer.

So among the four options only option (A) is correct.
Hence option (A) is the correct answer.

Note: Frequency of a wave is defined as the number of cycles of the waves crossing per unit time. Frequency is solemn property of the wave and is generally fixed, the physical definition of time period is the time after which the cycles of a wave are periodically repeated, for example the time period of sine wave is $2\pi$, that means after this interval the waves of sin periodically repeats itself. Wavelength is nothing but the distance travelled by the wave cycles in a unit time.