Parabolic Sag Defines String Equilibrium

The presence of a constant external force, such as gravity, fundamentally changes the string's baseline for oscillation, shifting it from the flat $u=0$ axis to a fixed parabolic equilibrium curve. This shift is derived mathematically by setting the time-dependent term to zero ($\partial_t^2 u=0$) in the wave equation, which results in Poisson's equation in one dimension: $\partial_x^2 u=\frac{\rho_{\ell} g}{S}$. The general solution to this equation is $u=\rho_{\ell} g x^2 / 2 S+A x+B$, demonstrating that the stationary profile of the string under tension is parabolic. Physically, this parabolic sag curve becomes the new resting position. Therefore, the total observed displacement is the result of the principle of superposition, where dynamic oscillation occurs relative to this gravity-induced parabolic equilibrium curve, rather than around a flat baseline.

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