Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 336 / 1 / b / i

Table of contents
- Solution i

Solution

Here

L_{1} = - c^{2} ϕ_{x}, L_{2} = ϕ_{t}, L_{3} = 0,

so the Euler-Lagrange field equation is the variable-coefficient wave equation

ϕ_{tt} - \partial_{x} (c^{2} ϕ_{x}) = 0.

At leading order,

Φ = A (X, T) cos θ

has

ϕ_{x} \sim k Φ_{θ}, ϕ_{t} \sim - ω Φ_{θ},

and hence

L^{(0)} = \frac{1}{2} (ω^{2} - c^{2} k^{2}) Φ_{θ}^{2} .

The leading part of the modulated-wave first integral is

\partial_{θ} [\frac{1}{2} (ω^{2} - c^{2} k^{2}) Φ_{θ}^{2}] = 0.

A nonconstant periodic wave therefore requires the local dispersion relation

ω^{2} = c^{2} (X, T) k^{2} .

This is usually called the eikonal equation, or equivalently the leading geometric-optics dispersion relation.

 Ancestors (11)

 View article source

 Discussion (0)

There are no discussions about this article yet.

 Articles by others on the same topic (0)

There are currently no matching articles.

  See all articles in the same topic Create my own version