A sinusoidal transverse wave of displacement amplitude Aᵢ is incident from string 1 onto string 2. The strings are under the same tension and have wave impedances Z₁ and Z₂. Which pair correctly gives the reflected displacement amplitude Aᵣ and transmitted displacement amplitude Aₜ, including the sign of Aᵣ?
Options

Question

A sinusoidal transverse wave of displacement amplitude Aᵢ is incident from string 1 onto string 2. The strings are under the same tension and have wave impedances Z₁ and Z₂. Which pair correctly gives the reflected displacement amplitude Aᵣ and transmitted displacement amplitude Aₜ, including the sign of Aᵣ?

Options

G1 Physics · Accepted Answer

Let the junction be at x = 0. Continuity of displacement gives Aᵢ+Aᵣ = Aₜ. Continuity of transverse force gives k₁(Aᵢ−Aᵣ) = k₂Aₜ. For strings under the same tension, k ∝ Z. Solving the two equations gives Aᵣ/Aᵢ = (Z₁−Z₂)/(Z₁+Z₂) and Aₜ/Aᵢ = 2Z₁/(Z₁+Z₂). Therefore, Aᵣ = [(Z₁−Z₂)/(Z₁+Z₂)]Aᵢ, Aₜ = [2Z₁/(Z₁+Z₂)]Aᵢ. Correct option: B.