Bounds on complex polarizabilities and a new perspective on scattering by a lossy inclusion

Here, we obtain explicit formulas for bounds on the complex electrical polarizability at a given frequency of an inclusion with known volume that follow directly from the quasistatic bounds of Bergman and Milton on the effective complex dielectric constant of a two-phase medium. We also describe how analogous bounds on the orientationally averaged bulk and shear polarizabilities at a given frequency can be obtained from bounds on the effective complex bulk and shear moduli of a two-phase medium obtained by Milton, Gibiansky, and Berryman, using the quasistatic variational principles of Cherkaev and Gibiansky. We also show how the polarizability problem and the acoustic scattering problem can both be reformulated in an abstract setting as “$Y$ problems.” In the acoustic scattering context, to avoid explicit introduction of the Sommerfeld radiation condition, we introduce auxiliary fields at infinity and an appropriate “constitutive law” there, which forces the Sommerfeld radiation condition to hold. As a consequence, we obtain minimization variational principles for acoustic scattering that can be used to obtain bounds on the complex backwards scattering amplitude. Some explicit elementary bounds are given.

Figure 1

The dashed lines show bounds on $\alpha /h^2$ representing the orientationally averaged real and complex polarizability per unit volume of an arbitrarily shaped two-dimensional inclusion. In (a) the dielectric constant ${\epsilon }_1$ of the inclusion is real, while in (b) it is purely imaginary. The surrounding medium has a dielectric constant of unity. In (b) “ru” and “rl” denote the upper and lower bounds on the real part of $\alpha /h^2$, while “iu” and “il” denote the upper and lower bounds on the imaginary part of $\alpha /h^2$. The solid lines are the numerical results for a square-shaped inclusion. The bounds in (a) are asymptotically attained in the cases of the solid circular cylinder and the thin cylindrical shell. The figures are reproductions, with permission of Springer, of Figs. 2 and 3 in [6].