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Two waves of same intensity I₀ emitted from two sources having same phase difference (φ). Due to superposition of two waves, the intensity of resultant wave is directly proportional to ______ .

Asked in GUJCET 2025 · Intensity and coherent sources

• (1) sin²(φ/2)
• (2) sin²φ
• (3) cos²(φ/2)
• (4) cos²φ

Step-by-step solution

Why the other options are wrong

• (1) Follows from a sign slip in the interference term, I=2I₀(1-cos φ). That expression vanishes at φ=0, where two waves arriving in step must instead give the greatest possible intensity, 4I₀.
• (2) Keeps the whole angle where the half angle belongs, and carries the sign slip as well. It would make the resultant vanish at φ=0 and at φ=π alike, although those are the brightest and the darkest cases.
• (4) The right shape with the wrong argument: the half angle comes from the identity 1+cos φ=2 cos²φ/2. As a test, cos²φ would be greatest at φ=π, where two equal waves exactly out of step must cancel.

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