Question

Let the velocity of a wave be $ {{V}} $ , time period $ {{T}} $ , frequency $ {{\gamma }} $ and wavelength $ {{\lambda }} $ . Mark the correct relation between these three.
(A) $ {\text{VT}} = \dfrac{1}{{{\lambda }}} $
(B) $ {\text{VT}} = {{\lambda }} $
(C) $ \dfrac{{\text{V}}}{{\text{T}}} = {{\lambda }} $
(D) $ \dfrac{{\text{T}}}{{\text{V}}} = {{\lambda }} $

Answer 1

Hint : To solve this question, we need to look into the basic definition of each of the characteristics of the wave given in the question. Then correlating them with the basic formula of the velocity of a particle, we can get the required relation.

Formula used: The formula which is used to solve this question is given by
 $ v = \dfrac{d}{t} $ , here $ v $ is the velocity of a particle which travels a distance of $ d $ in the time $ t $ .

Complete step by step answer
The variables which are given in the question are characteristics of a wave. For deducing the relation between these, we have to look into their basic definitions.
Wavelength $ {{\lambda }} $ :
The wavelength of a wave is defined as the distance between two successive crests or troughs.
Time period $ {\text{T}} $ :
The time period of a wave is defined as the time required for the wave to travel a distance equal to its wavelength $ {{\lambda }} $ .
Frequency $ {{\gamma }} $ :
The frequency of a wave is defined as the number of oscillations completed by the wave in one unit of time. So it is equal to the inverse of the time period $ {\text{T}} $ of the wave.
Velocity $ {\text{V}} $ :
The velocity of a wave is defined as the distance travelled by the wave in one unit of time.
Now, we know that the formula of the velocity is given by
 $ v = \dfrac{d}{t} $
Looking at the definitions of the wavelength $ {{\lambda }} $ , the time period, and the velocity of the wave from above, we distance $ d $ is equivalent to the wavelength, the time $ t $ is equivalent to the time period $ {\text{T}} $ , and the velocity $ v $ is equivalent to the velocity of the wave $ {\text{V}} $ . Therefore the relation between these three characteristics of the wave is given by
 $ {\text{V}} = \dfrac{{{\lambda }}}{{\text{T}}} $
Multiplying by the time period $ {\text{T}} $ on both the sides we finally get
 $ {\text{VT}} = {{\lambda }} $
Hence, the correct answer is option B.

Note
The frequency given in the question is not useful in this solution. This is because it is not there in any of the four options given in the question. So we do not need to use the frequency in our answer.