The equations of two waves are given by :
y₁ = 5 sin 2π(x - vt) cm
y₂ = 3 sin 2π(x - vt + 1.5) cm
These waves are simultaneously passing through a string. The amplitude of the resulting wave is :

Solution

Physics sir · 3 steps

Find the phase difference first. Here it is 3π, so the two waves fight each other and amplitudes subtract.

1. Phase difference. The two phases are 2π(x - vt) and 2π(x - vt) + 2π × 1.5. They differ by φ = 3π, which is an odd multiple of π, so cos φ = -1.

   A = √(A₁² + A₂² + 2A₁A₂cos φ)

2. Put in numbers. With A₁ = 5 cm, A₂ = 3 cm and cos 3π = -1: A = √(25 + 9 - 30) = √4 = 2 cm. This is the same as A₁ - A₂, because the waves are exactly out of phase.
3. So. The resulting wave has amplitude 2 cm, which is option (a).

The idea to keep. When two waves of the same frequency superpose, the phase difference decides everything. An even multiple of π means amplitudes add, and an odd multiple of π means they subtract.

Answer from the NTA final answer key. Solution by PrepWiser.