Properties of Standing Waves on a String

Steps:

• Nodes in a standing wave are fixed points of zero displacement.
• The amplitude at any point depends on its distance from the nearest node, following the sinusoidal pattern.
• All points on the string oscillate at the same frequency due to the wave's uniform driving.
• Points equidistant from a node lie symmetrically, sharing the same amplitude magnitude.

Why D is correct:

• In standing waves, amplitude is given by $2A |\sin(kx)|$, where nodes occur at $\sin(kx) = 0$; equidistant points from a node yield equal $|\sin|$ values, ensuring identical amplitudes, with frequency uniform across the string.

Why the others are wrong:

• A: Points across a node vibrate 180° out of phase due to sign change in the sine term.
• B: While frequency is the same, phases differ by 180° between adjacent antinodes.
• C: Equidistant points from N1 have equal amplitudes but opposite phases (180° shift).

Final answer: D