Forskarskolan i tvärvetenskaplig matematik

The potential method for Radar Cross Section calculation of Perfectly Electrically Conducting surfaces

Supervisor: Magnus Herberthson (MAI)

For bodies with perfectly conducting surfaces (PEC), the problem of determining the the radar cross section (RCS) simplifies to an two dimensional integral equation over the surface. Namely, if a PEC body is illuminated with an e.g. plane wave, it will induce a surface current $\mathbf J$ which determines the scattered field and therefore also the radar cross section. $\mathbf J$ is determined by the condition that the scattered field $\mathbf E_s$ at the surface, $S$ say, has a tangential component which annihilates the tangential component of the incoming electric field.

In the frequency domain, (and with a natural choice of coordinate axes), the equation is $$ \forall \mathbf{r} \in S: E_0 e^{-ikz}\hat x \, \hat =\, i kc\mu_0 (\mathbf{I}+\frac{1}{k2} \nabla \nabla \cdot)\int_S g(\mathbf{r},\mathbf{r}')\mathbf{J}(\mathbf{r}') dS' $$ Here $g(\mathbf{r},\mathbf{r}')$ is the Green's function $\displaystyle\frac{e^{i k |\mathbf{r}-\mathbf{r}'|}}{4 \pi |\mathbf{r}-\mathbf{r}'|}$, while $E_0, k, c, \mu_0$ are various constants. The symbol $\hat =$ stands for equality of the tangential parts.

The project considers the following two facts:

• By multiplying both sides of the above equation with $e^{ikz}$ and regarding it as an equality of forms on $S$, the LHS is closed.
• When $S$ is homeomorphic to a sphere, the surface current $\mathbf J$ can be decomposed as $\mathbf{J}=d'\Phi+\delta'\!*\!\Psi$ In practise, this decomosition may be applied to $\mathbf{K(\mathbf{r}')}=e^{ikz'}\mathbf J(\mathbf{r}')$.

The aim of the project is to investigate the implications of these facts, both practical numerical aspects and theoretical aspects.