Two waves are simultaneously passing through a string and their equations are :
y₁ = A₁ sin k(x - vt), y₂ = A₂ sin k(x - vt + x₀). Given amplitudes A₁ = 12 mm and A₂ = 5 mm, x₀ = 3.5 cm and wave number k = 6.28 cm⁻¹. The amplitude of resulting wave will be ................ mm.

Solution

Physics sir · 3 steps

Turn the shift x₀ into a phase: 3.5 wavelengths puts the two waves in opposite phase.

1. Find the phase difference. The two waves differ only by the extra term kx₀ inside the sine. So the phase difference is φ = kx₀ = 6.28 × 3.5 = 2π × 3.5 = 7π rad.
2. Read the phase. The wavelength is λ = 2π/k = 1 cm, so x₀ is 3.5 wavelengths. The extra half wavelength puts the waves in opposite phase, and cos 7π = -1.

   A = √(A₁² + A₂² + 2A₁A₂cos φ) = √(144 + 25 - 120) = √49

3. So. The resultant amplitude is |A₁ - A₂| = 12 - 5 = 7 mm.

The common slip: 17. Treating 3.5 wavelengths like a whole number of wavelengths makes the waves look in phase, and adding the amplitudes gives 17 mm instead of 7.

The idea to keep. For two waves of the same k and v, phase difference is k × path shift. Whole wavelengths do not matter; only the leftover fraction decides between adding and subtracting amplitudes.

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