Wave period and point displacement at half-period

Steps:

• Frequency f = 0.5 Hz, so period T = 1/f = 2 s.
• Time t = 1 s equals T/2.
• Point X on the rope undergoes simple harmonic motion with period T.
• At t = T/2, displacement of X is the negative of its initial displacement y(0), inverting the position vertically.

Why B is correct:

• B shows the inverted position of X, matching the half-period phase shift where y(t) = -y(0) for SHM in transverse waves.

Why the others are wrong:

• A: Matches original position, correct for t = 0, not t = 1 s.
• C: Shows quarter-period advance (upward max if original at equilibrium), incorrect for half-period.
• D: Shows three-quarter-period advance (downward max), not half-period inversion.

Final answer: B