Constraint condition on transformed relation for generalized acoustics

Abstract

Contrary to transformation optics (TO), there exist many possibilities for transformed relations of material property and field variable in case of transformation acoustics (TA). To investigate the underlining mechanism and develop a general method that can obtain the full transformed relations, an alternative interpretation to the form-invariance is explored. We consider a spatial transformation, with which a physical phenomenon described in an initial space is transformed to a deformed space, and interpret the mapping by local affine transformation point-by-point. Further, we postulate that the transformed material property and field must rebuild the same physical process, and that the energy must be conserved at each point during the transformation. These conditions impose the constraint on the transformed relation for material property and field. By establishing two local Cartesian frames defined uniquely by the spatial transformation, any physical quantity is shown to first experience a rigid rotation and then a stretch operation during the transformation. We show that the constraint conditions are not enough to determine completely the transformed relation for TA, leaving a possibility to define them differently as found in the literature. New acoustic transformations with constant density or modulus are also proposed and verified by constructing a two-dimensional acoustic cloak. Finally, we show that the transformed relation is uniquely determined for transformation optics, and discuss how this method can be extended to other transformation physics.