From 5th November to 21st November I took part in the modelling course run by Professor John Ockendon FRS . I was part of a group looking at the mathematics of clocaking an object. Our project leader was Andrew Duncan MSc MSc and the other members of are team were Amal Alphonse MMath MSc and Maria Veretennikova MMath MSc. We focused on modelling cloaking in one dimension focusing on waves that obeyed the Helmholtz equation

$$ \phi'' (x) +k^2 \phi(x) =0 ~~~~k \in \mathbb R $$

except in a region from \( 0\) to \( 1\) where it obeys a modified Helmholtz equation

$$ \phi''(x) +k(x)^2 \phi(x) =0 $$

with the boundary conditions

$$ \phi(0)=1 ,~ \phi'(0)= ik_0, ~ \phi(1)=0$$

This means that we have a wave coming from the left which enters a region at \(0 \) ,where it is continuous and its derivative is continuous across the boundary, and vanishes by the end of the region at \(1\).

The form of \(k (x)\) we focused on is

$$ k(x) = (1-x) ^{-n} $$

we prove there is no solution for \(n <1 \) and analyse a method of regularised cloaking (i.e. \( k(x+\epsilon) \) ) for \(n>1\). Regularised cloaking has the advantage that \(k (x) < \infty \) which is more likely to be physically obtainable. The full results of our work can be found here and the slides of our talk summarising the results can be found here .