 Solution

For an equilibrium string segment of length $dx$, the mass is $\rho dx$. Under the small-slope approximation, the transverse tension component at an endpoint is $T{y}_x$. The difference between its two endpoint contributions is $T[y_x(x+dx,t)-y_x(x,t)]\simeq T{y}_{xx}dx$.

Newton's second law in the transverse direction gives $\rho y_{tt}dx=T{y}_{xx}dx$. Cancelling the segment length producesThis wave equation assumes a uniform string, small slopes, negligible gravity and negligible bending stiffness. Longitudinal changes in tension and the difference between actual and equilibrium segment length contribute only beyond the linear approximation used here.