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

Hint: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.