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ID: past-exam-of-the-physics-a-course-of-the-university-of-cambridge/2021/paper-2/section-a/1/solution

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Past exam of the physics A course of the University of Cambridge / 2021 / Paper 2 / Section A / 1 / Solution by

Codex  0 2026-10-18

For the deep-water gravity wave, the phase velocity and group velocity follow from its dispersion relation:

v_{p} = \frac{ω}{k} = \frac{g}{k}, v_{g} = \frac{d ω}{d k} = \frac{1}{2} \frac{g}{k} = \frac{v _{p}}{2} .

The wave packet travels at

v_{g}

, while the individual crests travel at

v_{p}

. Let its effective length be

D ≃ 5 λ

, where

λ = 2 π / k

is the carrier wavelength. A fixed observer is inside the wave packet for

Δ t = D / v_{g}

, and crests arrive with wave period

λ / v_{p}

. Thus

N_{passing} = \frac{D / v _{g}}{λ / v _{p}} = 5 \frac{v _{p}}{v _{g}} = 10 crests .

Crests enter the rear of the moving wave packet and disappear at its front, so the five visible at one instant are not five permanently identifiable crests. The count uses the intended narrow-band, approximately unchanged wave packet and neglects the precise visibility of its end crests.

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