Wave Equation Sand Castle Mold

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What pairs perfectly with the beach than waves? And what pairs better with waves than the WAVE EQUATION, you ask? We approximated the iconic sand castle mold with a circle of 2-D Gaussian functions centered 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 several time points to capture the different mold shapes. Now, let's dive into interpreting these shapes. Imagine stretching a drumhead into the shape of a castle (like in the first mold) and then releasing it. The shapes of these molds are actually a visual representation of vibrations in the membrane as time progresses. You can even start to see them rebounding back onto the drum in the last mold. Check out a stunning GIF of the sandcastle's dramatic downfall here and explore the math behind it in more detail here!