Wave-2

\begin{align} P &= \vec{\tau}\cdot \vec{v_{y}}\ \\ &= \tau \frac{\partial y}{\partial t}\sin\theta \\ &\simeq - \tau \frac{\partial y}{\partial t} \frac{\partial y}{\partial x} \\ &=\tau\omega ky_{m}^2 \cos^2(kx-\omega t) \\ \end{align}

\begin{align} \implies P_{\text{avg}}&=\frac{1}{2}\tau\omega ky_{m}^2 \\ &=\frac{1}{2} \mu v^2 \omega ky_{m}^2 \\ &=\frac{1}{2} \mu v\omega^2 y_{m}^2 \end{align}