Wave-2

\begin{aligned} P &= \boldsymbol{\tau}\cdot \mathbf{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{aligned}

\begin{aligned} \implies P_{\mathrm{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{aligned}