 Side localization of modified Helmholtz plane waves

On a square, a normalized modified Helmholtz adjoint plane wave with positive normal growth toward one side has magnitude one there, magnitude $e^{-2\eta }$ on the opposite side, and an adjacent-side magnitude integral bounded by $1/\eta$. Thus large tangential spectral frequencies suppress remote-side normal-trace coefficients. Paired sine collocation of square modified Helmholtz global relations further cancels adjacent-side unknown traces exactly.