Question

Two coherent waves are represented by \( y_1 = a_1 \cos \omega t \) and \( y_2 = a_2 \sin \omega t \), and they are superimposed on each other. The resulting intensity is proportional to:

• A. \( (a_1 + a_2)^2 \)
• B. \( (a_1 - a_2)^2 \)
• C. \( a_1^2 + a_2^2 \)
• D. \( \sqrt{a_1^2 + a_2^2} \)

Hint:

When two coherent waves are superimposed, the intensity is proportional to the square of the resultant amplitude, which depends on the square of the individual amplitudes.

Answer 1

The Correct Option is C

The intensity \( I \) of a wave is proportional to the square of the amplitude of the wave. When two waves \( y_1 \) and \( y_2 \) are superimposed, their resultant displacement \( y_{\text{result}} \) is the sum of the individual displacements. Thus: \[ y_{\text{result}} = y_1 + y_2 = a_1 \cos \omega t + a_2 \sin \omega t \] The intensity \( I \) is proportional to the square of the amplitude: \[ I \propto y_{\text{result}}^2 = (a_1 \cos \omega t + a_2 \sin \omega t)^2 \] Expanding this: \[ I \propto a_1^2 + a_2^2 + 2a_1a_2 \cos(\omega t)\sin(\omega t) \] Since the intensity is proportional to the square of the amplitude, the correct expression is: \[ I \propto a_1^2 + a_2^2 \] Thus, the correct answer is option (C).