Question

Derive the relation between the velocity of wave, wavelength and frequency.

Answer 1

Hint:
Wave length is the distance travelled by the wave during the time a particle of the medium completes one vibration. Using the above definition of the wave, try to derive the relation between the velocity of wave, wavelength and frequency.

Complete answer:
 We know that the Wavelength is the distance travelled by the wave during the time a particle of the medium completes one vibration.
Let $\lambda$ be the wavelength, $v$ be the velocity and $T$ be the time period
We know that the velocity is defined as the distance travelled per time taken.
Mathematically it can be represented as $v={\displaystyle \frac{d}{t}}$
It implies $v={\displaystyle \frac{\lambda }{T}}$ since the distance travelled is $\lambda$ and the time taken is $T$
We know that the frequency is inversely proportional to the time period.
Mathematically it can be represented as $\vartheta ={\displaystyle \frac{1}{T}}$
Put this in above relation, we obtain the final relation as
$v=\vartheta \lambda$
Therefore the wave velocity = frequency $\times$ wavelength is the relation between the velocity of wave, wavelength and frequency.

Note:
Under the same physical conditions the wave velocity in a medium remains constant. When the wave travels from one medium to another the frequency never changes and when the wave travels into the denser medium the wavelength decreases and it slows down.