🎬how the string's equilibrium is fundamentally shifted by the constant external force

This animation demonstrates the principle of superposition in the solution to the inhomogeneous wave equation, where a constant external force (like gravity) is present. The total displacement of the string, $u(x, t)$ , is shown as the sum of two parts: a stationary solution $u_{\text {stat }}(x)$ (the fixed parabolic sag curve caused by gravity) and a dynamic oscillating solution $u_{\text {osc }}(x, t)$ (the fundamental standing wave). Crucially, the string does not oscillate around the flat $u=0$ axis; instead, the dynamic motion occurs relative to the parabolic equilibrium curve, which is the new resting position defined by the constant force. The visualization clearly tracks the instantaneous sag point at the string's center, illustrating how the gravitational field acts to permanently offset the string's baseline displacement.

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how the string's equilibrium is fundamentally shifted by the constant external force | Animated Resultsviadean on Notion