Question

What is the phase difference between two successive crests in the wave ?
A) $\pi$
B) $\dfrac{\pi }{2}$
C) $2\pi$
D) $4\pi$

Answer 1

The path difference between the two successive crest of a wave is called the wavelength of the wave ($\lambda$). Using the relationship between the phase difference and the path difference $\Delta \varphi = \dfrac{{2\pi }}{\lambda } \times \Delta x$, and putting the value of path-difference ( $\Delta x = \lambda$) and further simplifying we get the required result.

Step by step solution :-
Phase difference between two successive crests in the wave determines the harmony in the vibrating state of two particles; it means how far one point on the wave is lagging or leading the others.

There is a relationship of path difference and the phase difference of the wave and it is given by
Phase difference = $\dfrac{{2\pi }}{{wavelength}}times\;path\;difference$.
Phase difference = $\Delta \varphi$
Path difference = $\Delta x$
Wavelength ($\lambda$) - The path difference between the two successive crest of a wave is called the wavelength of the wave ($\lambda$).
$\Rightarrow \Delta \varphi = \dfrac{{2\pi }}{\lambda } \times \Delta x$……………. equation (1)
Path difference between two successive crests is $\lambda$.
It means
$\Rightarrow \Delta x = \lambda$
Putting the value of the path difference in equation (1), we get
$\Rightarrow \Delta \varphi = \dfrac{{2\pi }}{\lambda } \times \lambda$
$\Rightarrow \Delta \varphi = 2\pi$
Since, the phase difference between two successive crests in the wave is $2\pi$.

Hence, option ( C ) is the correct answer.

Note:- Since phases are the angles , any whole full turns should usually be ignored when performing arithmetic operations on them.
Phases can also be defined as “ The relative displacement of two waves with respect to each other ” .
While phase difference of a wave can be defined as “The time interval by which a wave leads by or lags by another wave” and the phase difference is not a property of only one wave, it’s the relative property of two or more waves which we are dealing with. This is also called phase angle or phase offset.