Past exam of the mathematics course of the University of Cambridge / 2019 / ii / Paper 4 / 38A / a

Table of contents
- Solution a

Solution

The physical rapid phase is

Θ (x, t) = \frac{θ ( x , t )}{ε} .

Define the local wavevector, angular frequency, and group velocity by

k_{i} = \frac{\partial Θ}{\partial x _{i}} = \frac{1}{ε} \frac{\partial θ}{\partial x _{i}}, ω = - \frac{\partial Θ}{\partial t} = - \frac{1}{ε} \frac{\partial θ}{\partial t}, c_{g, i} = \frac{\partial Ω}{\partial k _{i}} .

The equality of mixed partial derivatives gives the phase-compatibility equations

\frac{\partial k _{i}}{\partial t} + \frac{\partial ω}{\partial x _{i}} = 0, \frac{\partial k _{i}}{\partial x _{j}} = \frac{\partial k _{j}}{\partial x _{i}} .

A ray is defined to move with the local group velocity:

\frac{d x _{i}}{d t} = \frac{\partial Ω}{\partial k _{i}} .

Differentiate the local dispersion relation

ω = Ω (k; x, t)

with respect to

x_{i}

. Along a ray,

\frac{d k _{i}}{d t} = \frac{\partial k _{i}}{\partial t} + \frac{d x _{j}}{d t} \frac{\partial k _{i}}{\partial x _{j}} = - \frac{\partial ω}{\partial x _{i}} + \frac{\partial Ω}{\partial k _{j}} \frac{\partial k _{i}}{\partial x _{j}} = - \frac{\partial Ω}{\partial x _{i}} - \frac{\partial Ω}{\partial k _{j}} \frac{\partial k _{j}}{\partial x _{i}} + \frac{\partial Ω}{\partial k _{j}} \frac{\partial k _{i}}{\partial x _{j}} = - \frac{\partial Ω}{\partial x _{i}} .

The last two terms cancel because

k

is a gradient. Similarly,

\frac{d ω}{d t} = \frac{\partial Ω}{\partial t} + \frac{\partial Ω}{\partial x _{i}} \frac{d x _{i}}{d t} + \frac{\partial Ω}{\partial k _{i}} \frac{d k _{i}}{d t} = \frac{\partial Ω}{\partial t} .

Finally,

\frac{1}{ε} \frac{d θ}{d t} = \frac{\partial Θ}{\partial t} + \frac{d x _{j}}{d t} \frac{\partial Θ}{\partial x _{j}} = - ω + k_{j} \frac{\partial Ω}{\partial k _{j}} .

Thus the Hamiltonian ray-tracing equations are

\frac{d x _{i}}{d t} = \frac{\partial Ω}{\partial k _{i}}, \frac{d ω}{d t} = \frac{\partial Ω}{\partial t}, \frac{d k _{i}}{d t} = - \frac{\partial Ω}{\partial x _{i}}, \frac{1}{ε} \frac{d θ}{d t} = - ω + k_{j} \frac{\partial Ω}{\partial k _{j}} .

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