A sinusoidal wave is propagating along a stretched string that lies along the x-axis. The displacement of the string as a function of time is graphed in figure for particles at x=0 and x=0.09 m. These two points are with in one wave length of each other. The speed of the wave, if the wave is moving in positive x-direction. [Take $\sin^{- 1} (\frac{3}{4}) = 0.589$ (radians)]

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3 m/s

9.1 m/s

3.8 m/s

2.2 m/s

answer is B.

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Detailed Solution

$y = 4 \sin (ωt - kx + ϕ)$

at t=0 and x=0, y=0 $\Rightarrow ϕ = 0$

at t=0 and x=0.09, y=3

$\begin{array}{l} \Rightarrow 3 = 4 \sin (0 - k (0.09)) \\ \Rightarrow \frac{3}{4} = \sin [- k (0.09)] \end{array}$

Case 1 :

$\begin{array}{l} \Rightarrow \frac{3}{4} = \sin (K (0.09) + π) = 0.589 = K (0.09) + π \\ \Rightarrow K = \frac{0.589 - 3.142}{0.09} = - 28.367 \end{array}$

This is not possible as K cannot be negative.

Case 2 :

$\begin{array}{l} \frac{3}{4} = \sin (K (0.09) - π) = 0.589 = (0.09) K - 3.142 \\ \Rightarrow K = \frac{3.142 + 0.589}{0.09} = 41.556 rad / m \end{array}$