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The (1D) wave equation is $$\frac{\partial^2y}{\partial t^2} = v^2 \frac{\partial^2y}{\partial x^2},$$ where for simplicity let's just assume $v$ is a constant independent of time $t$ or space $x$. This is a differential equation describing a function $y$ of $t$ and $x$. A solution to the wave equation is any expression $y(x,t)$ such that differentiating ...
That is the heat equation in polar coordinates with axial symmetry. The (isotropic) heat equation without sources or sinks is $$\frac{\partial U}{\partial t} - K\nabla^2U =0.$$ If you look up the Laplacian operator in cylindrical coordinates, you will find that your expression matches this exactly.