String Wave Equation Derivation

Total upward force on length $dx$ string element:

$$\begin{eqnarray*} f(x+dx/2) &=& K\sin(\theta_1) + K\sin(\theta_2)\\ &\approx& K... ...\approx& K\left[-y'(x) + y'(x)+y''(x)dx)\right]\\ &=& Ky''(x)dx \end{eqnarray*}$$

Mass of length $dx$ string segment: $m=\epsilon \,dx$.

By Newton's law, $f=ma=m{\ddot y}$, we have

$$Ky''(t,x)dx = (\epsilon \,dx){\ddot y}(t,x)$$

or

$$\zbox{Ky''(t,x) = \epsilon {\ddot y}(t,x)}$$