What goes better with the beach than waves? And what goes better with waves than the WAVE EQUATION??
We approximated the classic sand castle mold with a circle of 2-D Gaussian functions around a bump function. Setting this as our initial condition and imposing a Dirichlet boundary condition, we then simulated the wave equation PDE in Mathematica and sampled at a couple time points to get the different mold shapes.
How can we interpret these shapes? Imagine stretching a drumhead into the shape of a castle (like in the first mold) and then letting it go. The shapes of these molds are what vibrations in the membrane would look like as time evolves. You can even start to see them rebound into the drum in the last mold.