Curl completion of a transverse oscillatory wave

Core idea

Let $a:U\to\mathbb C^3$ and $\Phi:U\to\mathbb R$ be smooth, with $n=\nabla\Phi\ne0$ , $n\cdot a=0$ , and real $\kappa\ne0$ . Define

W=\nabla\times\left(\frac{i}{\kappa}\frac{n\times a}{|n|^2}e^{i\kappa\Phi}\right).

Then $\nabla\cdot W=0$ , and

W=e^{i\kappa\Phi}\left(a+ \frac{i}{\kappa}\nabla\times\frac{n\times a}{|n|^2}\right).

Cross products with complex amplitudes use the complex bilinear extension of the real cross product. This is a curl completion of the transverse principal amplitude $a$ .

Verification and size of the correction

Use $\nabla\times(b e^{i\kappa\Phi})=e^{i\kappa\Phi}(\nabla\times b+i\kappa n\times b)$ and $n\times(n\times a)=-|n|^2a$ . Divergence of a curl vanishes. With a positive lower bound for $|n|$ and bounds on derivatives of $n,a$ , the displayed correction coefficient has an explicit factor $\kappa^{-1}$ . Derivatives of the full corrected wave also differentiate the exponential and need separate estimates. Smoothly supported vector potentials preserve support under this construction.