Particle velocity in transverse waves peaks at maximum slope in the snapshot

Steps:

• The transverse particle velocity is given by $v_p = -v_w \frac{\partial y}{\partial x}$, where $v_w$ is wave speed.
• Maximum $|v_p|$ occurs where the absolute slope $|\frac{\partial y}{\partial x}|$ is greatest.
• For a right-moving wave, the position with the steepest negative slope has particles moving upward, aligning with the wave's forward propagation phase.
• Point A marks this steepest slope location on the graph.

Why A is correct:

• A is at the inflection point with maximum $|\frac{\partial y}{\partial x}|$, yielding fastest particle speed per the wave equation.

Why the others are wrong:

• B: Near crest; slope near zero, so low particle velocity.
• C: On gentler slope; moderate $|\frac{\partial y}{\partial x}|$, slower than maximum.
• D: Near trough; slope smaller than at A, reducing speed.

Final answer: A