A transverse wave is represented by the equation y=y0 sin(2π/λ)(vt-x) For what value of λ, is the maximum particle velocity equal to two times the wave velocity?

Question

A transverse wave is represented by the equation y=y0 sin(2π/λ)(vt-x) For what value of λ, is the maximum particle velocity equal to two times the wave velocity? (A) λ=(π y0/2) (B) λ=(π y0/3) (C) λ=2π y0 (D) λ=π y0

Tardigrade Admin · Accepted Answer

The given equation of wave is y=y0 sin (2 π/λ)(v t-x) Particle velocity =(d y/d t)=y0 cos (2 π/λ)(v t-x) ⋅ (2 π v/λ) ((d y/d t)) max =y0 ⋅ (2 π/λ) v ∴ y0 ⋅ (2 π/λ) v=2 v or, λ=π y0

Q. A transverse wave is represented by the equation $y = y_{0} sin \frac{2 π}{λ} (v t - x)$ For what value of $λ$ , is the maximum particle velocity equal to two times the wave velocity?

$λ = \frac{π y _{0}}{2}$

$λ = \frac{π y _{0}}{3}$

$λ = 2 π y_{0}$

$λ = π y_{0}$

The given equation of wave is
$y = y_{0} sin \frac{2 π}{λ} (v t - x)$
Particle velocity $= \frac{d y}{d t} = y_{0} cos \frac{2 π}{λ} (v t - x) \cdot \frac{2 π v}{λ}$
$(\frac{d y}{d t})_{m a x} = y_{0} \cdot \frac{2 π}{λ} v$
$∴ y_{0} \cdot \frac{2 π}{λ} v = 2 v$
or, $λ = π y_{0}$

Q. A transverse wave is represented by the equation y=y0​sinλ2π​(vt−x) For what value of λ, is the maximum particle velocity equal to two times the wave velocity?

λ=2πy0​​

λ=3πy0​​

λ=2πy0​

λ=πy0​