Many-body wave scattering problems are solved asymptotically, as the size $a$ of the particles tends to zero and the number of the particles tends to infinity. Acoustic, quantum-mechanical, and electromagnetic wave scattering by many small particles is studied. This theory allows one to give a recipe for creating materials with a desired refraction coefficient. One can create material with negative refraction, that is, the group velocity in this material is directed opposite to the phase velocity. One can create material with some desired wave-focusing properties. For example, one can create a new material which scatters plane wave mostly in a fixed given solid angle.

REFERENCES:

0) A.G.Ramm, Scattering of Acoustic and Electromagnetic Waves by Small Bodies of Arbitrary Shapes. Applications to Creating New Engineered Materials, Momentum Press, New York, 2013

1) A.G.Ramm, Wave scattering by small bodies of arbitrary shapes, World Sci. Publishers, Singapore, 2005. 2) A.G.Ramm, Inverse problems, Springer, New York, 2005.

3) A.G.Ramm, Many-body wave scattering by small bodies and applications, J. Math. Phys., 48, N10, 103511, (2007).

4) A.G.Ramm, Wave scattering by many small particles embedded in a medium, Phys. Lett. A, 372/17, (2008), 3064-3070.

5) A.G.Ramm, Electromagnetic wave scattering by small bodies, Phys. Lett. A, 372/23, (2008), 4298-4306.

6) A.G.Ramm, Creating desired potentials by embedding small inhomogeneities, J. Math. Phys., 50, N12, 123525, (2009).

7) A.G.Ramm, A method for creating materials with a desired refraction coefficient, Internat. Journ. Mod. Phys B, 24, 27, (2010), 5261-5268.

8) A.G.Ramm, Materials with a desired refraction coefficient can be created by embedding small particles into a given material, International Journal of Structural Changes in Solids (IJSCS), 2, N2, (2010), 17-23.

9) A.G.Ramm, Scattering of scalar waves by many small particles, AIP Advances, 1, 022135, (2011).

10) A.G.Ramm, Scattering of electromagnetic waves by many thin cylinders, Results in Physics, 1, N1, (2011), 13-16.

11) A.G.Ramm, Distribution of particles which produces a "smart" material, Jour. Stat. Phys., 127, N5, (2007), 915-934.

12) A.G.Ramm, Wave scattering by many small bodies and creating materials with a desired refraction coefficient, Afrika Matematika, 22, N1, (2011), 33-55.

13) A.G.Ramm, Many-body wave scattering in the case of small scatterers, J. Appl. Math. Comput., (JAMC), 41, N1, (2013), 473-500.

14) A.G.Ramm, Electromagnetic wave scattering by small impedance particles of an arbitrary shape, J. Appl. Math. Comput., (JAMC), 43, N1, (2013), 427-444.

15) M.Andriychuk and A.G.Ramm, Numerical solution of many-body wave scattering problem for small particles and creating materials with desired refraction coefficient, Chapter in the book: "Numerical Simulations of Physical and Engineering Processes", InTech., Vienna, 2011, pp.1-28. (edited by Jan Awrejcewicz)