Absorbent Strip in a Hard Baffle Wall, as a Variational Problem

The geometry and field composition are as in → Sect.  D.10 .

The variational principle is based on Helmholtz's theorem of superposition, which requires that the power of the “cross intensity” $p(x_{0},0)\cdot v_{z}^{a}(x_{0},0)$ of the desired field $p(x,z)$ and the particle velocity $v_{z}^{a}(x,z)$ of the adjoint field be zero at the plane $z=0$. The adjoint field is the solution for exchanged emission and immission points. The cross power is minimised by variation of the amplitude P of the estimate $P\cdot\exp(-jkx\cdot\sin\Theta _{{i}})$. The expression to be minimised is

$$\begin{array}[]{@{}l@{\,}l@{\,}l}\displaystyle 4P_{i}ka\, G\cdot P&-&\displaystyle kaG\cdot P^{2}-\frac{1}{2}\, k^{2}G^{2}P^{2}\int\limits _{{-a/2}}^{{+a/2}}{e^{{jkx\cdot\sin\Theta _{i}}}dx}\\ &&\cdot\displaystyle\int\limits _{{-a/2}}^{{a/2}}{H_{0}^{{(2)}}(k|x-x_{0}|)\cdot e^{{-j\, kx_{0}\cdot\sin\,\Theta _{i}}}\, dx_{0}}=Min(P)\;.\\ \end{array}$$...