Shadow response of a Schwarzschild black hole in a gravitational standing wave

Motivated by Mandel [Phys.Rev. D 72 (2005) 084025] and Szybka et al. [Phys. Rev. D 100, 064025 (2019)], we develop a perturbative theory of the shadow response of a Schwarzschild black hole to an auxiliary weak standing-wave perturbation. The shadow is defined geometrically as the capture-escape separatrix of the null Hamiltonian flow on an observer's screen. We use the standing field as a time-symmetric boundary-value approximation rather than as a generic equilibrium state of an isolated black hole. The physical radiative field is obtained by adding the radiation-reaction sector, which restores outgoing behavior at future null infinity and downgoing behavior at the future horizon. The standing-wave sector is characterized covariantly by an effective high-frequency stress tensor with equal counter-propagating null fluxes. We derive a first-order transfer functional from the metric perturbation to the displacement of the critical curve. In a fixed observer-screen and areal-radius representative, the leading response is set by the contraction of the perturbation with the tangent to the unstable photon-sphere congruence at $r=3M$. A Regge-Wheeler-Zerilli decomposition yields mode-resolved transfer coefficients and parity selection rules. Reflection-symmetric polar modes stretch or squeeze the shadow, whereas reflection-antisymmetric modes generate leading centroid shifts. The linear transfer functional also maps the standing and radiation-reaction sectors to the physical critical curve. Damped quasinormal forcing gives a coherent decaying modulation, while persistent standing forcing can split the photon-sphere separatrix and generate chaotic capture-escape structure.