= Fixed-frequency nonradiating source
{title2=$Q=(\Delta+k^2)\chi,\qquad \chi\in C_c^\infty(A)$}
A source is nonradiating at frequency $k$ if its outgoing field has zero <far-field pattern>. For every $\chi\in C_c^\infty(A)$, the source $Q=(\Delta+k^2)\chi$ has outgoing field $\psi=\chi$ and is invisible outside $A$. Integration by parts shows that its <Fourier transform> vanishes on $|\boldsymbol\xi|=k$, so it belongs to the <kernel of a linear map> defined by the <source-to-far-field operator at fixed frequency>. This family proves nonuniqueness independently of the instability of the inverse.
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