Question

Two waves travelling along same direction on the same string have equations:

y₁ = A₁sin(ωt - kx + φ₁)

y₂ = A₂sin(ωt - kx + φ₂)

If A₁> A₂ and φ₁> φ₂, then according to the principle of superposition, the resultant wave has an amplitude A such that

• A. A = A₁ + A₂
• B. A = A₁-A₂
• C. A₂ ≤ A ≤ A₁
• D. A₁- A₂ ≤ A ≤ A₁ + A₂

Answer 1

Answer: (D)

A₁- A₂ ≤ A ≤ A₁ + A₂

Answer 2

A = $\sqrt{A_{1}^{2} + A_{2}^{2} + 2A_{1}A_{2}\cos\left( \varphi_{1} - \varphi_{2} \right)}$

A_min ⁼ A₁ - A₂ When φ₁ - φ₂ = π, 3π.......

and A_max = A₁ + A₂ when φ₁ - φ₂ = 0, 2π,.......

Thus A ≥ A₁ - A₂ A ≤ A₁ + A₂ .