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Past exam of the mathematics course of the University of Cambridge / 2017 / iii / Paper 342 / 1 / c / Solution by

Codex  0 Created 2026-10-03 Updated 2026-10-06

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t = 0

when gravity is switched off. The subsequent overdamped relaxation of a clamped--free filament obeys

ζ_{⊥} h_{t} = - B h_{xxxx}

with the same homogeneous endpoint conditions. Separate a normal mode as

h = ϕ (x / L) e^{- Γ t}

s = x / L

gives

ϕ^{''''} = α^{4} ϕ, Γ = \frac{B α ^{4}}{ζ _{⊥} L ^{4}} .

The two clamp conditions reduce the general hyperbolic/trigonometric solution to

ϕ (s) = C [cosh (α s) - cos (α s)] + D [sinh (α s) - sin (α s)] .

The two free-end conditions have matrix

(cosh α + cos α sinh α - sin α sinh α + sin α cosh α + cos α) (C D) = 0.

Its determinant is

2 (1 + cosh α cos α)

. Hence the clamped--free bending mode values satisfy

cos α_{n} cosh α_{n} = - 1, α_{1} = 1.8751040687 \dots, α_{2} = 4.6940911330 \dots .

There is no zero mode: at zero eigenvalue

ϕ

is cubic, and the four clamp/free conditions force all coefficients to vanish. With the following normalization a mode is

ϕ_{n} (s) = cosh (α_{n} s) - cos (α_{n} s) - η_{n} [sinh (α_{n} s) - sin (α_{n} s)], η_{n} = \frac{c o s h α _{n} + c o s α _{n}}{s i n h α _{n} + s i n α _{n}} .

The fourth-order operator is self-adjoint and positive under these endpoint conditions, since

\int f g^{''''} = \int f^{''} g^{''}

. Its orthogonal eigenfunction expansion gives

h (x, t) = \sum_{n \geq 1} c_{n} ϕ_{n} (x / L) exp [- \frac{B α _{n}^{4} t}{ζ _{⊥} L ^{4}}], c_{n} = \frac{\int _{0}^{L} h ( x , 0 ) ϕ _{n} ( x / L ) d x}{\int _{0}^{L} ϕ _{n} ( x / L ) ^{2} d x} .

For the previously attained steady profile, the first coefficient is nonzero:

h_{s}

is positive and the first mode can be chosen positive in the interior. Two applications of integration by parts, using both functions' endpoint conditions, also give

c_{1} = \frac{w L ^{4}}{B α _{1}^{4}} \frac{\int _{0}^{1} ϕ _{1} ( s ) d s}{\int _{0}^{1} ϕ _{1} ( s ) ^{2} d s} .

Thus the dominant long-time answer, including its shape and amplitude, is

h (x, t) \sim c_{1} ϕ_{1} (x / L) e^{- t / τ_{1}}, τ_{1} = \frac{ζ _{⊥} L ^{4}}{B α _{1}^{4}} = \frac{ζ _{⊥} L ^{4}}{12.36236337 \dots B} .

The next mode decays much faster. For other initial shapes with an accidentally zero first-mode projection, the lowest mode with

c_{n} \neq = 0

determines the asymptotic behaviour. The decay time has the correct units and exhibits the strong

L^{4}

dependence characteristic of bending opposed by local viscous drag.

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