Question

if two waves with equal amplitudes and wavelengths travel through a medium in such a way that a particular particle of the medium is at the crest of one wave and at the trough of the other wave at the same time, what will happen to that particle?

Explanation:

Step1: Recall wave interference concept

Wave interference occurs when two waves meet. Constructive interference is when crests (or troughs) align, increasing amplitude. Destructive interference is when a crest of one wave aligns with a trough of another.

Step2: Analyze given conditions

The two waves have equal amplitudes and wavelengths. A particle is at the crest of one wave and the trough of the other simultaneously. This is destructive interference.

Step3: Determine the particle's motion

For destructive interference with equal amplitudes, the displacements of the two waves at the particle's position are equal in magnitude but opposite in direction (e.g., if one wave has a displacement of \( +A \) (crest), the other has \( -A \) (trough), where \( A \) is amplitude). The resultant displacement is \( A + (-A) = 0 \). So the particle will undergo destructive interference and remain stationary (or have zero displacement) due to the cancellation of the two waves' effects.

Answer:

The particle will undergo destructive interference and remain stationary (its displacement will be zero) because the two waves' effects (crest and trough at the same point for the particle) cancel each other out (equal amplitude, opposite displacement at that point).