Question

For constructive interference the phase difference between two waves must be ___________________.
A. $0,\pi ,2\pi ,3\pi ,............$
B. $0,\pi ,2\pi ,3\pi ,............$
C. $0,2\pi ,4\pi ,6\pi ,............$
D. $0,3\pi ,6\pi ,9\pi ,............$

Answer 1

Constructive interference between two waves takes place only when the path difference is an integral multiple of $\lambda$.
The above statement can be written in the form of an equation as follows:
$\Delta x = n\lambda$ where n is any integer (i.e. $n = 0,1,2,3,.......$)

Phase Difference is given by the formula:
$= > \delta = \dfrac{{2\pi }}{\lambda }\Delta x$
Using the above two equations, we can easily compute phase difference for constructive interference.

Complete step by step solution:
Constructive interference between two waves takes place only when the path difference is an integral multiple of $\lambda$.
The above statement can be written in the form of an equation as follows:
$\Delta x = n\lambda$ where n is any integer (i.e. $n = 0,1,2,3,.......$)
Now, Phase Difference is given by the formula:
$= > \delta = \dfrac{{2\pi }}{\lambda }\Delta x$
Inserting the value of $\Delta x$in the above equation,

We get,
$= > \delta = \dfrac{{2\pi }}{\lambda }n\lambda$
$= > \delta = 2\pi n$ where $n = 0,1,2,3,.......$
Therefore for constructive interference, phase difference must be $0,2\pi ,4\pi ,6\pi ,............$

Hence Option (C) is correct.

Note: We have mentioned that n can take any integral value ( technically, $n \in ( - \infty ,\infty )$) but still we have started n from 0 (i.e. we have not provided n with any negative values) Negative values have not been provided to n because we can clearly see in the options that all phase differences are positive (Hence we do not require negative phase differences).
Such questions require through conceptual understanding of the Waves Chapter. While not being a calculation intensive question, we can still observe the need to memorize all the formulas in order to solve this question (since this question was purely based on the application of formulas).