A compact exterior profile with a proved outgoing signal

This section supplies an analytical benchmark, not an evaluated experiment.

Choose any ${\omega }_0>0$. The two incoming generalized eigenfunctions are linearly independent, and $\tau ({\omega }_0)\ne 0$. Therefore

is not the zero solution. Choose a finite $x_0$ with $M_0=M(x_0,{\omega }_0)\ne 0$.

Let

Use normalized rescalings to choose real nonnegative bumps

with both integrals one. Choose $\delta$ small enough that

These finite choices exist by continuity.

Set

This has fixed compact exterior shell support.

For this profile,

where ${\hat{\chi }}_{\mathrm{plain}}(\omega )=\int {\chi }_{\delta }(t)e^{\mathrm{i}\omega t}\mathrm{d}t$. In particular $d_0=d_{\infty }({\omega }_0)\ne 0$.

Choose a compact positive-frequency band $B_0$ around ${\omega }_0$ on which

Let $H$ be a nonnegative normalized smooth bump inside $B_0$, and define

Then