A thick uniform rope of length L is hanging from a rigid support. A transverse wave of wavelength λ₀ is set up at the middle of rope as shown in figure. The wavelength of the wave as it reaches to the topmost point is

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$λ_{0}$

$\sqrt{2} λ_{0}$

$\frac{λ_{0}}{\sqrt{2}}$

$2 λ_{0}$

answer is B.

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Detailed Solution

Speed of pulse at a distance x from bottom, $v = \sqrt{gx}$

While traveling from mid point to the top, frequency remains unchanged.

$\frac{v_{1}}{λ_{1}} = \frac{v_{2}}{λ_{2}} \Rightarrow \frac{\sqrt{g (\frac{L}{2})}}{λ_{0}} = \frac{\sqrt{gL}}{λ_{2}} \Rightarrow λ_{2} = \sqrt{2} λ_{0}$

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