C.4 Standing waves and resonance

A node is a point on a standing wave where the displacement is always zero. It arises from complete destructive interference between the two superposed waves.

The two component waves have equal amplitude but travel in opposite directions. At nodal positions, their displacements are always equal and opposite, giving a resultant displacement of zero at all times.

The only statement that correctly identifies zero displacement and destructive interference is: "The rope remains stationary at all times due to complete destructive interference."

The question asks for the correct description of what occurs at a node of a standing wave.

"Maximum amplitude due to constructive interference" describes an antinode, not a node. This option is incorrect.

"Oscillates with half the amplitude" is not a feature of nodes or antinodes. The amplitude at nodes is zero, not half the original amplitude.

"Oscillates in phase with every other point" is a partial truth about antinodes in the same segment but does not describe what happens at a node.

Option B correctly states that the rope remains stationary due to complete destructive interference, which is the defining property of a node.

In a standing wave, all points between two consecutive nodes oscillate in phase with each other. Points in adjacent segments (separated by one node) are in antiphase.

Consecutive nodes are separated by $\frac{\lambda }{2}$. Two points separated by $\frac{\lambda }{2}$ are at the boundaries of the same or adjacent half-wavelength loops. Specifically, two antinodes separated by $\frac{\lambda }{2}$ lie in different segments and are in antiphase. However, two general points separated by $\frac{\lambda }{2}$ can be in different loops.

Actually, two points separated by exactly $\frac{\lambda }{2}$ span exactly one loop of the standing wave. Points within the same loop are in phase. Points in adjacent loops are in antiphase. Two points separated by $\frac{\lambda }{2}$ sit at symmetrically equivalent positions in adjacent half-loops, meaning they are in phase with each other (e.g., two antinodes that are both at maximum positive or maximum negative displacement simultaneously).

Two points separated by $\frac{\lambda }{2}$ in a standing wave oscillate in phase with each other, matching option A.

This question tests knowledge of phase relationships between points on a standing wave separated by $\frac{\lambda }{2}$.

"Separated by one node" is a spatial description, not a phase relationship. It does not answer what their oscillation relationship is.

Antiphase ($\pi$ rad difference) applies to points in adjacent segments separated by an odd multiple of $\frac{\lambda }{2}$ that cross a node between them. Two points in the same half-loop separated by $\frac{\lambda }{2}$ are actually in phase, not antiphase.

"Both are nodes" would only be true if the two points happen to fall exactly on nodes, which is not generally the case for any arbitrary pair of points separated by $\frac{\lambda }{2}$.

Option A is correct: two points separated by $\frac{\lambda }{2}$ on a standing wave oscillate in phase with each other.