JEE Main 2025 Apr 7 Shift 2, Physics Q13: Types and Equations of Waves

The transverse velocity is $v_y=10\pi \cos (20\pi x+10\pi t)$, so two points $x_1$ and $x_2$ have the same oscillating speed at all times when $|\cos (20\pi x_1+10\pi t)|=|\cos (20\pi x_2+10\pi t)|$ holds identically in $t$.

This is satisfied either when the two points are in phase (separation $=n\lambda$) or in anti-phase (separation $=\frac{n\lambda }{2}$ with $n$ odd); anti-phase points have exactly opposite transverse velocities, so their speeds match.

The wave number is $k=20\pi$ rad/m, so $\lambda =\frac{2\pi }{k}=\frac{1}{10}$ m $=10$ cm.

The minimum non-zero separation is $\frac{\lambda }{2}=5$ cm.