components. Plane-wave eigenmodes Another way to solve the one-dimensional wave equation is to first analyze its frequency eigenmodes. A so-called eigenmode is a solution that oscillates in time with a well-defined constant angular frequency , so that the temporal part of the wave function takes the form , and the amplitude is a function of the spatial variable , giving a separation of variables for the wave function: This produces an ordinary differential equation for the spatial part : Therefore, which is precisely an eigenvalue equation for , hence the name eigenmode. It has the well-known plane-wave solutions with wave number . The total wave function for this eigenmode is then the linear combination where complex numbers , depend in general on any initial and boundary conditions of the problem. Eigenmodes are useful in constructing a full solution to the wave equation, because each of them evolves in time trivially with the phase factor so that a full solution can be decomposed into an eigenmode expansion: or in terms of the plane waves, which is exactly in the same form as in the algebraic approach. Functions 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 , 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 . 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. Vectorial wave equation in three space dimensions The vectorial wave equation (from which the scalar wave equation can be directly derived) can be obtained by applying a force equilibrium to an infinitesimal volume element. In a homogeneous continuum (cartesian coordinate ) with a constant modulus of elasticity a vectorial, elastic deflection causes the stress tensor . The local equilibrium of a) the tension force due to deflection and b) the inertial force caused by the local acceleration can be written as By merging density and elasticity module the sound velocity results (material law). After insertion,