A one-dimensional phase-sum model for delay compensation and phase accumulation in reradiated light

Abstract When a transparent medium is represented as a collection of driven microscopic sources, students may ask why reradiated contributions can add coherently in the forward direction while a comparable coherent buildup does not generally occur in the backward direction. We present a one-dimensional classroom phase-sum model that distinguishes the driving wavenumber k_i, which prescribes the spatial phase of the sources, from the signed propagation wavenumber k_s of the observed reradiated component. Forward coherence corresponds to zero wavevector mismatch (k_s=k_i). In the equal-magnitude backward comparison (k_s=-k_i), a position-dependent phase remains, so the bulk contributions do not generally build coherently. The same relation is introduced first through a two-source steady-state phase-retardation argument and then extended to phasor sums. Continuous uniform and discrete disordered source distributions are distinguished because their backward residuals scale differently. The model is not a microscopic derivation of refractive index, Fresnel coefficients, or the Ewald-Oseen extinction theorem. Its contribution is pedagogical: a proposed sequence connecting elementary traveling-wave phase, two-source reasoning, phasor addition, and standard phase-matching ideas for introductory undergraduate optics.