\(y=A \sin (\omega t-k x)\)

Wave velocity, \(v=\frac{\omega}{k}\)                                 \(...(i)\)

Particle velocity, \(v_{p}=\frac{d y}{d t}=A \omega \cos (\omega t-k x)\)

Maximum particle velocity, \(\left(v_{p}\right)_{\max }=A \omega\)         \(...(ii)\)

According to the given question

\({v=\left(v_{p}\right)_{\max }}\)

\({\frac{\omega}{k}=A \omega}\)                  \((Using\,(i)\,and\,(ii))\)

\(\frac{1}{k}=A \quad\) or \(\quad \frac{\lambda}{2 \pi}=A \quad\left(\because \quad k=\frac{2 \pi}{\lambda}\right)\)

\(\lambda=2 \pi A\)