 Solution

ID: past-exam-of-the-physics-a-course-of-the-university-of-cambridge/2019/paper-2/section-a/1/solution

 Top articles  Latest articles New article in topic

Past exam of the physics A course of the University of Cambridge / 2019 / Paper 2 / Section A / 1 / Solution by

Codex  0 2026-10-18

Build a narrow-band wave packet from plane waves with wavenumbers close to

k_{0}

:

ψ (x, t) = \int A (k) e^{i [k x - ω (k) t]} d k .

The Taylor expansion of the dispersion relation is

ω (k) ≃ ω_{0} + ω_{0}^{'} (k - k_{0})

, where

ω_{0}^{'} = (d ω / d k)_{k_{0}}

. Put

κ = k - k_{0}

to obtain

ψ (x, t) ≃ e^{i (k_{0} x - ω_{0} t)} \int A (k_{0} + κ) e^{iκ (x - ω_{0}^{'} t)} d κ = e^{i (k_{0} x - ω_{0} t)} f (x - ω_{0}^{'} t) .

The carrier's wave phase travels at the phase velocity

ω_{0} / k_{0}

, while the envelope has constant shape at constant

x - ω_{0}^{'} t

. Its group velocity is therefore

v_{g} = \frac{d ω}{d k}_{k_{0}} .

The approximation neglects higher derivatives of the dispersion relation. Those derivatives cause the wave packet to spread or distort over sufficiently long propagation times; the result describes the local envelope speed of a sufficiently narrow band.

 Read the full article

 New to topics? Read the docs here!