Five waves of equal speed are superposed: y₁ = 8 sin(ωt + kx), y₂ = 6 sin(ωt + kx + π/2), y₃ = 4 sin(ωt + kx + π), y₄ = 2 sin(ωt + kx + 3π/2), and y₅ = 4√2 sin(ωt - kx + π/4). Find the resultant wave.
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Question

Five waves of equal speed are superposed: y₁ = 8 sin(ωt + kx), y₂ = 6 sin(ωt + kx + π/2), y₃ = 4 sin(ωt + kx + π), y₄ = 2 sin(ωt + kx + 3π/2), and y₅ = 4√2 sin(ωt - kx + π/4). Find the resultant wave.

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G1 Physics · Accepted Answer

Step 1: The first four waves have a common phase ωt+kx. Their phasor sum is (8−4)+i(6−2)=4+4i=4√2∠π/4. Step 2: They give 4√2 sin(ωt+kx+π/4). Adding y₅ = 4√2 sin(ωt−kx+π/4) gives 8√2 cos(kx) sin(ωt+π/4). Final: A.

Five waves of equal speed are superposed: y₁ = 8 sin(ωt + kx), y₂ = 6 sin(ωt + kx + π/2), y₃ = 4 sin(ωt + kx + π), y₄ = 2 sin(ωt + kx + 3π/2), and y₅ = 4√2 sin(ωt − kx + π/4). Find the resultant wave.