The demo uses a dynamic animation to illustrate the fundamental difference between ideal and realistic wave behavior by comparing the solutions to the Undamped and Damped wave equations. The left plot visualizes the Undamped Wave ( $\partial_t^2 u-c^2 \partial_x^2 u=0$ ), showing a standing wave oscillation that maintains a constant amplitude over time, representing an idealized system that perfectly conserves energy. In contrast, the right plot visualizes the Damped Wave ( $\partial_t^2 u+k \partial_t u-c^2 \partial_x^2 u=0$ ), where the inclusion of the damping term ( $+k \partial_t u$ ) causes the wave's amplitude to decay exponentially, visibly shrinking its oscillating envelope until the system eventually comes to rest, accurately modeling the inherent energy loss found in real-world systems like a string vibrating in air .
visually compares the behavior of an undamped wave and a damped wave over time-L.mp4
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