d
(d) On comparing the given equation with standard equation \(y = a\sin \frac{{2\pi }}{\lambda }(vt - x)\).

It is clear that wave speed \({(v)_{wave}} = v\) and maximum particle velocity

\({({v_{\max }})_{particle}} = a\omega = {y_0} \times \) co-efficient of \(t  = {y_0} \times \frac{{2\pi v}}{\lambda }\)

\({({v_{\max }})_{particle}} = 2 (\omega)_{wave}\) ==> \(\frac{{a \times 2\pi v}}{\lambda } = 2v\) ==> \(\lambda = \pi {y_0}\)