I have a function $u(r,t)$ that satisfies $$\frac{\partial^2u}{\partial t^2} - c^2 \Delta u = 0.$$
I'm looking at a solution of a problem with this setup and it states that the above equation can be written as
$$\frac{\partial^2u}{\partial t^2} + c^2Au = 0$$ with $$A = -\Delta = -\frac{1}{r}\frac{d}{dr}(r\frac{d}{dr}).$$ I have two questions:
The operator $\Delta$ is defined as $\sum \frac{\partial^2}{\partial x_i^2}$ but I never see the term $\frac{\partial^2}{\partial t^2}$ in $\Delta u$ when looking at solutions. Why is this?
I get $$Au = -\frac{1}{r}\frac{d}{dr}(r\frac{du}{dr}) = -\frac{1}{r}(ru')' = -u'' - \frac{u'}{r}$$ which is NOT equal to $-\Delta u = \frac{\partial }{\partial r^2}$ or $-\Delta u = \frac{\partial }{\partial r^2} + \frac{\partial }{\partial r^2}.$
What is going on?