First slide

Travelling wave

Question

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

Moderate

A

$λ = 2 {πy}_{0}$
B

$λ = \frac{2 π y_{0}}{3}$
C

$λ = \frac{π y_{0}}{3}$
D

$λ = {πy}_{0}$

Solution

Given $y = y_{0} \sin \{\frac{2 π}{λ} (v t - x)\}$
$ω = \frac{2 π}{λ} v, k = \frac{2 π}{λ}$
Velocity of wave $V_{ω} = v$
Velocity of particle $V_{P} = \frac{\partial y}{\partial t} = y_{0} \frac{2 π v}{λ} \cos \{\frac{2 π}{λ} (v t - x)\}$
Maximum velocity of particle $V_{\max} = \frac{2 π v y_{0}}{λ}$
given that $V_{\max} = 2 V_{w a v e}$
$∴ \frac{2 π v y_{0}}{λ} = 2 v$
$∴ λ = π y_{0}$
Therefore, the correct answer is ( D )

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