Pluck location determines string harmonics

The initial physical configuration of a plucked string is mathematically defined by the displacement function $f(x)$ when the string, fixed at its ends, is pulled transversally at a specific point $x=x_0$. The subsequent dynamics are governed by initial conditions, including being released from rest at time $t=t_0$ ($u_t\left(x, t_0\right)=0$), which is necessary for a unique solution to exist for the second-order wave equation. This sharp initial condition, described as the "V at the pluck point," mandates that the resulting complex periodic motion must be visualized as an ongoing demonstration of the principle of superposition, requiring the inclusion of a large number of high-frequency harmonic modes (standing waves) to accurately represent the initial waveform fidelity.

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