🎬visualize how an initial wave profile splits into two equal and opposite-traveling components

The demo provides a dynamic visualization of the d'Alembert solution for the one-dimensional wave equation under a specific initial condition. Specifically, the animation shows an initial, centrally located wave pulse (created by a Gaussian function) that is assumed to be released from rest ( $\partial_t u(x, 0)=0$ ). Due to this zero initial velocity, the initial disturbance instantaneously resolves itself into two identical halves of the original pulse. The visualization clearly demonstrates these two component waves traveling away from the origin in opposite directions-one right-traveling ( $g(x-c t)$ ) and one left-traveling ($g(x+c t)$)-both propagating at the constant wave speed $c$. This illustrates the fundamental principle that any solution to the 1D homogeneous wave equation is a superposition of two such persistent, traveling profiles.

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visualize how an initial wave profile splits into two equal and opposite-traveling components | Animated Resultsviadean on Notion