# Boundary conditions on wave equation

I am having trouble understanding the boundary conditions.

From the solutions, the first is that $D_1(0, t) = D_2(0, t)$ because the rope can't break at the junction.

The second is that $\dfrac{\partial D_1}{\partial x} D_1(0, t) = \dfrac{\partial D_2}{\partial x}(0, t)$. How can I interpret this physically? I'm not quite sure how to think about $\partial D /\partial x$.

-

Imagine if this assumption were to fail in the following way: $$\frac{\partial D_1}{\partial x}(0,t) = -1, \qquad \frac{\partial D_2}{\partial x}(0,t) = 1$$ Then near the origin, the rope would look like the function $f(x) = |x|$ does at the origin; there would be a "triangular kink" in the rope facing upward.