# Physical meaning of non differentiatiability of $y(t)$ at a point of an elastic medium

Consider two waves $y_1,y_2$ travelling in opposite directions with equations $$y_1(x,t) = A \sin(\omega t - kx) \\ y_2(x,t) = A \sin(\omega t + kx)$$ That create the following standing wave $$y_s(x,t) = y_1 + y_2 = 2A\cos(kx)\sin (\omega t)$$

Consider, now, a point $P$ at $x = x_0$ (without loss of generality let $x_0 > 0$) whose motion as a function of time is described as

$$y_P = y(x_0,t) = \left\{ \begin{array}{lr} A \sin(\omega t + kx_0) & : 0 < t < kx_0 / \omega \\ 2A\cos(kx_0)\sin (\omega t) & : t \ge kx_0 / \omega \end{array} \right.$$

It is easy to show that $y_P$ is everywhere continuous, however is not always differentiable at $t_0 = kx_0 / \omega$.

My question: In case the elastic medium is not massless then what is the velocity of $P$ at time $t= t_0$? What's the physical meaning of non-differentiability of $y_P$? (The medium has infinite length)