Case: Read the passage and answer the question.
A wave is represented by a variable y which gives the displacement of the particles of the medium in which the wave is travelling at any given instant of time. In the same context, consider two waves y1 and y2 which superpose such that y gives the resultant of the two waves.
If \( a_1 \) and \( a_2 \) are the amplitudes of the two waves, what is the ratio \( I_{max}/I_{min} \)?
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → \( \frac{(a_1+a_2)^2}{(a_1-a_2)^2} \)
By the superposition principle the resultant displacement is \( y=y_1+y_2 \), and the intensity is proportional to the square of the amplitude. Take \( a_1\gt a_2 \).
Where the two waves arrive in phase, the displacements add and the resultant amplitude is largest, \( a_1+a_2 \).
Where they arrive in opposite phase, one displacement subtracts from the other and the amplitude is smallest, \( a_1-a_2 \).
Squaring each amplitude gives the two intensities:
\( I_{\max}\propto(a_1+a_2)^2 \)
\( I_{\min}\propto(a_1-a_2)^2 \)
\( \displaystyle \frac{I_{\max}}{I_{\min}}=\frac{(a_1+a_2)^2}{(a_1-a_2)^2} \)
NCERT (2026-27):
Coherent and Incoherent Addition of Waves – “… Since the intensity is proportional to the square of the amplitude, the resultant intensity will be given by
\( I=4I_0 \)
where I0 represents the intensity produced by each one of the individual sources; I0 is proportional to a2.”