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To understand this further, we take as a starting point the notion that coordinates transformations stretch nonvector objects, including scalar fields, power flow lines and surfaces of constant phase, as they would be stretched if they were tied to space itself.
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Together, this acoustic analysis and the corresponding electromagnetic analysis (Pendry et al 2006) show how the curl, divergence and gradient operators maintain form under arbitrary coordinate transformations, opening the door to analyzing other wave systems built on these three vector operators.
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Poynting vector S are parallel, but after the plane wave has been distorted by a cloaking transformation (Pendry et al 2006), these vectors are no longer parallel in the transformed region and they thus do not transform the same way.
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Center for Metamaterials and Integrated Plasmonics and Department of Electrical and Computer Engineering, Duke University, Durham, NC 27708, USA2 Department of Electrical and Computer Engineering, North Carolina State University, Raleigh, NC 27695, USA3 Author to whom any correspondence should be addressed.
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Demonstrating the invariance through analogy with electromagnetics or by a general analysis of the scalar Helmholtz equation masks some of the physics of the transformation approach, particularly how vectors such as particle velocity and the pressure gradient change under transformation.
We demonstrate the computation of these material parameters in two specific cases and show that the change in velocity and pressure gradient vectors under a nonorthogonal coordinate transformation is precisely how these vectors must change from purely physical arguments.
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Through an analysis of how power flow and constant phase surfaces must transform for completely general waves, we show that the velocity vector in acoustics must transform in a different way than the E and H vectors in electromagnetics.
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This confirms the material parameter expressions found previously by analogy and also demonstrates how the velocity vector must transform to maintain invariance, which is in agreement with that found from our qualitative argument based on wave physics.