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 Incoming links: Linearly damped string

Past exam of the mathematics course of the University of Cambridge / 2022 / ib / Paper 1 / 13B / a / Solution Created 2026-09-24 Updated 2026-09-25

Take positive

y

vertically downward. For a short string element, the transverse tension resultant is

T y_{xx} d x

, while gravity and linear drag contribute

μg d x

and

- 2 k μ y_{t} d x

. Newton's second law gives

μ y_{tt} d x = T y_{xx} d x - 2 k μ y_{t} d x + μg d x .

Since

T = μ c^{2}

, the damped wave equation is

y_{tt} + 2 k y_{t} - c^{2} y_{xx} = g, y (0, t) = y (l, t) = 0.

Choosing upward displacement reverses the sign of the gravity term.

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