A derivation for the acoustic material parameters in transformation domains

The concept of transformation acoustics offers a new strategy to design devices with different functionalities, such as cloaks, concentrators and others. The material parameters of the transformed shell can be expressed in the form of a multiplication of Jacobian matrix. Here we present a direct and different approach to analytically determine the material parameters of transformation shell. The proof ensures that, with the specified material parameters, the field exterior to the shell will be the same as that of the original medium, as that the governing equation remains unchanged and that the impedance match conditions are exactly fulfilled along the boundary. We will present our algorithm for a general class of devices, with two or three dimensional configurations, described by general orthogonal coordinates. The derivation is valid for any continuous transformation function corresponding to cloaks, anti-cloaks, rotators, concentrators, etc. In this work we focus on the mathematical framework of acoustic wave. But the same approach can be applied to electromagnetic wave as well. Explicit formulae for the mass density tensor and bulk modulus are derived in terms of the transformation function.