Eigenmodes are useful in constructing a full solution to the wave equation, because each of them evolves in time trivially with the phase factor e − − i ω ω t , so that a full solution can be decomposed into an eigenmode expansion : u ( x , t ) = ∫ ∫ − − ∞ ∞ ∞ ∞ s ( ω ω ) u ω ω ( x , t ) d ω ω , or in terms of the plane waves, u ( x , t ) = ∫ ∫ − − ∞ ∞ ∞ ∞ s + ( ω ω ) e − − i ( k x + ω ω t ) d ω ω + ∫ ∫ − − ∞ ∞ ∞ ∞ s − − ( ω ω ) e i ( k x − − ω ω t ) d ω ω = ∫ ∫ − − ∞ ∞ ∞ ∞ s + ( ω ω ) e − − i k ( x + c t ) d ω ω + ∫ ∫ − − ∞ ∞ ∞ ∞ s − − ( ω ω ) e i k ( x − − c t ) d ω ω = F ( x − − c t ) + G ( x + c t ) , which is exactly in the same form as in the algebraic approach. Functions s ± (ω) are known as the Fourier component and are determined by initial and boundary conditions. This is a so-called frequency-domain method, alternative to direct time-domain propagations, such as FDTD method, of the wave packet u (x, t), which is complete for representing waves in absence of time dilations. Completeness of the Fourier expansion for representing waves in the presence of time dilations has been challenged by chirp wave solutions allowing for time variation of ω. The chirp wave solutions seem particularly implied by very large but previously inexplicable radar residuals in the flyby anomaly and differ from the sinusoidal solutions in being receivable at any distance only at proportionally shifted frequencies and time dilations, corresponding to past chirp states of the source.