🎬how the three components—the quasi-static response and the transient response and the steady-state f

how the three components—the quasi-static response and the transient response and the steady-state forced response combine to form the total solution

This demonstration clearly showcases the Principle of Superposition for solving inhomogeneous partial differential equations by decomposing the total string displacement, $u(x, t)$, into three functional components. The key mathematical strategy involves transforming the original inhomogeneous boundary condition problem into a simpler system featuring homogeneous boundary conditions. This transformation yields $u(x, t)=u_0+v_1+v_2$, where $u_0$ is the quasi-static component that instantly satisfies the moving boundary condition, $v_1$ is the steady-state forced oscillation that persists indefinitely as a standing wave, and $v_2$ is the transient response required to satisfy the initial conditions, which quickly decays to zero due to implicit damping. Ultimately, the total motion highlights that the string's long-term, observable behavior is determined solely by the sum of the quasi-static boundary shape ( $u_0$ ) and the stable forced wave $\left(v_1\right)$.

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how the three components—the quasi-static response and the transient response and the steady-state forced response combine to form the total solution | Animated Resultsviadean on Notion