While receiving less attention in the literature than electromagnetic cloaking, theoretical efforts to define and create acoustic cloaks based upon mimicking coordinate transformations through use of metamaterials is of interest. The present work extends recent analysis of Norris [Proc. R. Soc. London, Ser. A 464, 2411-2434 (2008)] by considering a range of cloaks, from those comprised of fluid layers which are isotropic in bulk moduli with anisotropic density to those having anisotropic bulk moduli and isotropic density. In all but pure inertial varieties, fluid layers comprising the cloaks are pentamode materials governed by a special scalar acoustic equation for pseudopressure derived by Norris. In most cases presented, material properties of the fluid/pentamode layers are based upon target values specified by continuously varying properties resulting from theoretical coordinate transformations geared to minimize scattered pressure limited by realistic goals. The present work analyzes such cloaks for the specific case of plane wave scattering from an acoustically hard sphere. An initial exploration of the parameter space defining such cloaks (for example, material properties of its constituent layers, and operating frequency) is undertaken with a view toward "optimal" design.
Abstract.Alternative expressions for calculating the prolate spheroidal radial functions of the second kind (c. £) and their first derivatives with respect to £ are shown to provide accurate values over wide parameter ranges where the traditional expressions fail to do so. The first alternative expression is obtained from the expansion of the product of (c, £) and the prolate spheroidal angular function of the first kind (c, rj) in a series of products of the corresponding spherical functions. A similar expression for the radial functions of the first kind was shown previously to provide accurate values for the prolate spheroidal radial functions of the first kind and their first derivatives over all (o) parameter ranges. The second alternative expression for Rml (c, £) involves an integral of the product of (c, rf) and a spherical Neumann function kernel. It provides accurate values when £ is near unity and I -mis not too large, even when c becomes large and traditional expressions fail. The improvement in accuracy using the alternative expressions is quantified and discussed.
Abstract.Alternative expressions for calculating the prolate spheroidal radial functions of the first kind R^] (c, £) and their first derivatives with respect to £ are shown to provide accurate values, even for low values of I -m where the traditional expressions provide increasingly inaccurate results as the size parameter c increases to large values. These expressions also converge in fewer terms than the traditional ones. They are obtained from the expansion of the product of R^n) (c, £) and the prolate spheroidal angular function of the first kind 5^ (c, rj) in a series of products of the corresponding spherical functions. King and Van Buren [12] had used this expansion previously in the derivation of a general addition theorem for spheroidal wave functions. The improvement in accuracy and convergence using the alternative expressions is quantified and discussed. Also, a method is described that avoids computer overflow and underflow problems in calculating i?^(c, £) and its first derivative.
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