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The equation of a wave is given as y = y₀ sin (2π/λ)(vt − x). Find the maximum particle velocity.

1. Maximum particle velocity = y₀ × 2πv
2. Maximum particle velocity = y₀ × 2πλ
3. Maximum particle velocity = y₀ × λ/2π
4. Maximum particle velocity = y₀ × v/2π

Correct answer: Maximum particle velocity = y₀ × 2πv

Solution

The particle velocity is given by the time derivative of the displacement: v_particle = ∂y/∂t. Differentiating y = y₀ sin(2π/λ)(vt − x) with respect to t gives v_particle = y₀ × (2πv/λ) cos(2π/λ)(vt − x). The maximum value of cos is 1, so the maximum particle velocity is y₀ × 2πv.

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