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The singularity expansion method (SEM) represents transient scattering by superposition of damped oscillatory fields corresponding to the complex resonant frequencies of the scatterer. The series of these global wave fields, which encompass the scattering object as a whole, is slowly convergent at early observation times and even deficient at very early times when portions of the object are as yet unexcited. Thus, the resonance series representation must generally be augmented by an entire function in the complex frequency domain. The choice of the entire function is relatively arbitrary but affects the excitation coefficients, called coupling coefficients, of individual resonances and also the "turn-on" and "switch-on" times of the SEM series. Moreover, it contains essential (intrinsic) and nonessential (removable) portions which have been subjected to various interpretations. By formulating the transient problem in terms of traveling (progressing) incident, reflected and diffracted wavefronts, these constructs in the SEM can be interpreted in a precise and physical manner. Furthermore, the analysis clarifies the evolution of resonances as collective summations of multiple wavefront fields which are caused by successive reflections or diffractions at the surfaces and scattering centers comprising the object. By combining wavefronts and resonances self-consistently, one may construct a hybrid field that avoids the difficulties at early times in the SEM formulation. The systematic exploration of the interplay between wavefronts and resonances is facilitated through use of a flow diagram, as introduced in system theory. These concepts are developed in broad generality and are illustrated for two-dimensional scattering by various special configurations.