Knowlpedia

Definition

Transverse polarization of a vector wave

An amplitude lying in the plane perpendicular to the local wavevector.

A vector wave aeiκΦae^{i\kappa\Phi} is transversely polarized for κ≠0\kappa\ne0 at a point with ∇Φ≠0\nabla\Phi\ne0 if

∇Φ⋅a=0.\nabla\Phi\cdot a=0.

Its lies in the plane perpendicular to the phase gradient. For complex amplitudes the dot product with this real normal is extended complex linearly, so both real and imaginary parts are transverse.

Leading and exact incompressibility

The identity

∇⋅(aeiκΦ)=eiκΦ(∇⋅a+iκ∇Φ⋅a)\nabla\cdot(ae^{i\kappa\Phi}) =e^{i\kappa\Phi}\bigl(\nabla\cdot a+i\kappa\nabla\Phi\cdot a\bigr)

shows that transversality cancels the leading frequency term. Exact divergence freedom additionally requires ∇⋅a=0\nabla\cdot a=0, or a correction that cancels it. For a constant-amplitude plane wave transversality is sufficient.