We get $a = Y_0, \, \omega  = 2\ pi  f, k = \frac{{2\pi }}{\lambda }$.

Hence maximum particle velocity ${({v_{\max }})_{particle}} = a\omega = {Y_0} \times 2\pi f$

and wave velocity ${(v)_{wave}} = \frac{\omega }{k} = \frac{{2\pi f}}{{2\pi /\lambda }} = f\lambda $

$\because \,\,\,{({v_{\max }})_{Particle}} = 4{v_{Wave}}$==> ${Y_0} \times 2\pi f = 4f\lambda $ ==> $\lambda = \frac{{\pi {Y_0}}}{2}$.