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(B) In a plane progressive transverse wave,particles of the medium oscillate simple harmonically about their mean positions. Hence,the concepts of $SH

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$\lambda = \pi A/2$

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(B) In a plane progressive transverse wave,particles of the medium oscillate simple harmonically about their mean positions. Hence,the concepts of $SHM$ are applicable to the particles.The maximum particle velocity is given by $v_{p,max} = A \omega$,which occurs at the mean position.From the given wave equation:$y = A \sin (2 \pi f t - 2 \pi x / \lambda)$Comparing this with the standard wave equation $y = A \sin (\omega t - kx)$,we get:Angular frequency $\omega = 2 \pi f$Wave number $k = 2 \pi / \lambda$The wave velocity $v$ is given by:$v = \omega / k = (2 \pi f) / (2 \pi / \lambda) = \lambda f$According to the given condition,the maximum particle velocity is four times the wave velocity:$A \omega = 4 v$$A (2 \pi f) = 4 (\lambda f)$$2 \pi A f = 4 \lambda f$$\lambda = (2 \pi A) / 4$$\lambda = \pi A / 2$

$A$ transverse wave is described by the equation $y = A \sin [2\pi (f t - x/\lambda ) ]$. The maximum particle velocity is equal to four times the wave velocity if:

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