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The equation is a evolution of vibrations in a disk. For example a music instrument drum has a membrane fixed on a disk. The boundary condition says that the amplitude on the perimeter of the disk is 0, so it is fixed on a pipe. When you hit or apply force to the membrane you set it to an initial condition and after taking away the force the membrane starts to vibrate according to the equation. At the beginning of the solution it is assumed that the solution is a product of two functions: one does only depend on position and the other only on time. and this is indeed true, as we come to some solution.