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.

🎬Narrated Video

https://youtu.be/yBvyKzFjIJI

🏗️Structural clarification of Condensed Notes

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🗒️Downloadable Files - Recursive updates

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Copyright Notice

All content and images on this page are the property of Sayako Dean, unless otherwise stated. They are protected by United States and international copyright laws. Any unauthorized use, reproduction, or distribution is strictly prohibited. For permission requests, please contact [email protected]

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