Particle oscillation in transverse wave follows simple harmonic motion

Steps:

• At t=0, the snapshot graph gives the initial displacement y_0 of particle X at its fixed position x.
• In a transverse wave, each particle executes SHM with amplitude A (wave amplitude) and period T, so angular frequency ω = 2π/T.
• The phase φ is set by the wave equation y(x,t) = A sin(kx - ωt + φ), fixed for X's x, matching y_0 at t=0.
• The correct time graph shows sinusoidal oscillation starting at y_0 with period T.

Why C is correct:

• Matches SHM with initial displacement y_0 and period T per the wave equation for fixed x.

Why the others are wrong:

• A: Starts at zero displacement, mismatching y_0 from snapshot.
• B: Incorrect phase, not aligning with wave propagation direction.
• D: Non-sinusoidal or wrong period, violating SHM definition.

Final answer: C