Physics - Wave on String Question with Solution | TestHub

Let $u=\omega t-kx$, so $y=f(u)$. Then $\frac{\partial y}{\partial t}=\omega f^{\prime }(u)$ and $\frac{\partial y}{\partial x}=-k{f}^{\prime }(u)$, giving $\frac{\partial y}{\partial t}=-\frac{\omega }{k}\frac{\partial y}{\partial x}$. Differentiating again, $\frac{{\partial }^{2}y}{\partial t^2}={\omega }^2{f}^{\prime \prime }(u)$ and $\frac{{\partial }^{2}y}{\partial x^2}=k^2{f}^{\prime \prime }(u)$, so $\frac{{\partial }^{2}y}{\partial t^2}=\frac{{\omega }^2}{k^2}\frac{{\partial }^{2}y}{\partial x^2}$.