Using y₁=A sin(kx−ωt) and y₂=A sin(kx+ωt), with k=2π/λ and ω=2π/T, sketch both travelling profiles and their sum over 0≤x≤λ at t=0, T/4 and T/2. Mark nodes and antinodes. Explain why the flat resultant at T/4 does not mean the wave has disappeared or that all points are nodes.

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Resultants are 2A sin kx, 0 and −2A sin kx. Nodes at 0,λ/2,λ; antinodes at λ/4,3λ/4. At T/4 the opposite travelling profiles cancel in displacement only at that instant; non-nodal particles are moving through equilibrium. At t0 draw both A sin kx and sum 2A sin kx. At T/4 draw −A cos kx and +A cos kx, sum0. At T/2 draw both −A sin kx and sum −2A sin kx. Mark fixed nodes and maximum-amplitude antinodes correctly. Explain permanent zero amplitude versus temporary zero displacement. State that opposite travelling waves remain and particles at non-nodes pass through equilibrium.

Draw the waves and their sum over a cycle

Using y₁=A sin(kx−ωt) and y₂=A sin(kx+ωt), with k=2π/λ and ω=2π/T, sketch both travelling profiles and their sum over 0≤x≤λ at t=0, T/4 and T/2. Mark nodes and antinodes. Explain why the flat resultant at T/4 does not mean the wave has disappeared or that all points are nodes.

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