 Solution

For a short string element $dx$, small slope makes its tension magnitude approximately constant and its vertical component approximately $T{y}_x$. The net vertical force is $T[y_x(x+dx,t)-y_x(x,t)]\simeq T{y}_{xx}dx$. Its mass is $\rho dx$. Newton's second law givesThis assumes uniform equilibrium tension, small slopes, negligible bending stiffness and dissipation. The symbol $\rho$ is linear mass density here, not volume density.