Oscillatory modulation

The oscillatory modulation of a coefficient $a$ by a phase $\Phi$ is $a e^{i\kappa\Phi}$ . For smooth $a,\Phi$ , the product and chain rules give

\partial_j(ae^{i\kappa\Phi}) =e^{i\kappa\Phi}(\partial_ja+i\kappa a\partial_j\Phi).

Thus estimates on the amplitude alone do not control derivatives of the full wave independently of $\kappa$ .

Second derivatives

Componentwise for a vector amplitude,

\Delta(ae^{i\kappa\Phi})=e^{i\kappa\Phi} \left(\Delta a+2i\kappa\nabla\Phi\cdot\nabla a +i\kappa(\Delta\Phi)a-\kappa^2|\nabla\Phi|^2a\right).

For example, $\varepsilon\sin(x/\varepsilon)$ is small in value but has order-one first derivative. Frequency powers and amplitude powers must both be included when estimating a modulated field.