Phase difference in standing waves determines node-antinode separation

Steps:

• Standing wave displacement: $y(x,t) = 2A \sin(kx) \cos(\omega t)$, where $k = 2\pi / \lambda$.
• Phase difference between points X and Y is given by $k \Delta x$; 180° out of phase means $k \Delta x = \pi$.
• Solve for distance: $\Delta x = \pi / k = \lambda / 2$.
• At maximum displacement ($\cos(\omega t) = \pm 1$), X and Y reach antinodal peaks oppositely, confirming separation.

Why B is correct:

• In standing waves, 180° phase difference arises from $\sin(k(x + \lambda/2)) = -\sin(kx)$, per the wave equation's spatial term.

Why the others are wrong:

• A: Quarter wavelength gives 90° phase shift, not 180°.
• C: One wavelength gives 360° (0°) phase shift, in phase.
• D: Two wavelengths gives 720° (0°) phase shift, in phase.

Final answer: B