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Wave Speed expression

A wave pulse ...

Question

A wave pulse is generated in a string that lies along x-axis. At the points A and B are as shown in figure, if $R_{A} a n d R_{B}$ are ratio of wave speed to the particle speed respectively then

A

R_{A} > R_{B}

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B

R_{B} > R_{A}

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C

R_{A} = R_{B}

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D

Information is not sufficient to decide

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Solution

The correct option is B $R_{B} > R_{A}$
Wave speed is always constant at every point in the wave.
$∴ V_{A} w = V_{B} w$
Let the displacement of the particle be
$y = A sin (w t - k x) V_{p} = A cos (w t - k x) \times w x_{B} > x_{A} \Rightarrow w t - k x_{B} < w t - k x_{A} \Rightarrow cos (w t - k x_{B}) > cos (w t - k x_{A}) \Rightarrow V_{B} P > V_{A} P \frac{R_{A}}{R_{B}} = \frac{V_{A} w / V_{A} P}{V_{B} w / V_{B} P} \frac{R_{A}}{R_{B}} = \frac{V_{B} P}{V_{A} P} ∴ R_{A} > R_{B}$
Hence option (A) is correct.

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