\[\Delta E_P=\frac{1}{2}k(\Delta y)^2=\frac{1}{2}\frac{ES}{\Delta x}(\Delta y)^2=\frac{1}{2}ES\Delta x\left(\frac{\partial y}{\partial x}\right)^2\]
依杨氏模量\(E\)的定义和胡克定律，有
\[F=ES\frac{\Delta y}{\Delta x}=k\Delta y\]
又\(\Delta V=S\Delta x\)，\(u=\sqrt{\frac{E}{\rho}}\)，\(\frac{\partial y}{\partial x}=\frac{A\omega}{u}\sin\left[\omega\left(t-\frac{x}{u}\right)+\varphi\right]\)
\[\therefore\ \Delta E_P=\frac{1}{2}\rho A^2\omega^2\sin^2\left[\omega\left(t-\frac{x}{u}\right)+\varphi\right]\Delta V\]