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Progressing and oscillatory waves provide alternative building blocks for constructing source-excited time-harmonic or transient fields in various propagation and scattering environments. Progressing waves describe the field in terms of direct and multiple reflected-diffracted wavefront or ray arrivals, while oscillatory waves describe the field in terms of body resonances and (or) guided modes. Each description is convenient and physically incisive when it requires few constituent elements but inconvenient and physically more obscure when it requires many elements. Collective summation of many inconvenient elements into fewer convenient ones, when possible, provides a means of switching from a poorly to a more rapidly convergent field representation. Rays (wavefronts) and modes (resonances) fall into the category of such bilaterally convertible wave fields. They not only have complementary convergence properties but furnish, respectively, local and global environmental discriminants. When combined in self-consistent hybrid form, where the number retained of the one uniquely determines the grouping required of the other, there emerges a rigorous theory of propagation and diffraction with high versatility. Groupings can be chosen so as to eliminate "difficult" elements (for example, caustic forming or transitional ray fields) and replace these by "well behaved" ones (modes) of the complementary species, or to model multiple interference in the one as simple interference in the other. The theory, based on Poisson summation or on alternative treatment of wave spectral representations, is presented and applied to numerous coordinate separable, but also weakly nonseparable, environments in electromagnetics, underwater acoustics and elastic motion, with numerical comparisons to highlight the salient features of the hybrid appraoch.