Gauge invariance approach to acoustic fields

Abstract

We extend the gauge invariance property of the Maxwell's equations to the acoustic field equations. We use the analogy of the sound velocity as equivalent to the magnetic field and the acoustic stress field as equivalent to the electric field. We apply to sound propagation in solids and negative refraction. For sound propagation in solids, we use the equation of motion and the strain-displacement relation and express the velocity field and the stress field in terms of the vector potential and the scalar potential. We solve the inhomogeneous wave equation for these potentials and obtain the velocity field and the stress field in terms of these potentials. This gives a more rigorous solution than that from the Christoffel equation. Gauge invariance has symmetry property. We obtain the gauge form for the acoustic scalar potential and the vector potential and demonstrate their invariant property. We also show that the inhomogeneous wave equation in terms of the vector potential and the scalar potential possesses symmetry property. We also show the symmetry property of the acoustic field equation. We also derive the acoustic Lorentz condition from the equation of continuity. For the application of gauge invariance to negative refraction, we first show that the acoustic field equations also possess left hand and right hand symmetry and this gives rise to negative refraction. This is a different approach from that of Veselago who derives negative refraction from the negative permittivity and negative permeability. Here we extend negative refraction to anisotropic materials. In anisotropic materials, the compliance and stiffness possess rotational symmetry. When rotating in the clockwise direction, it gives rise to lefthanded phenomenon such as negative refraction. When rotating in the anticlockwise direction, it gives rise to the righthanded phenomenon such as positive refraction. Due to symmetry, both the righthanded phenomenon and the lefthanded phenomenon satisfy the acoustic field equations. The stress field, the velocity field and the acoustic Poynting vector together form a righthanded triplet or a lefthanded triplet depending on the direction of energy flow or the direction of the Poynting vector. According to parity conservation, acoustic law at the deepest level, there is no differentiation of righthanded and lefthanded treatment. The performance of an object and that of its mirror image will satisfy the same law of physics. The negative refraction in fact is a mirror image of the positive refraction.