Spectral spacetime-geometry of Womersley flow

We revisit the classical Womersley solution for pulsatile viscous flow in a circular tube and reconstruct its full time-domain geometry from first principles. By combining harmonic decomposition with exact Bessel solutions, we derive a unified spectral spacetime analytical solution in which the instantaneous relationship between pressure gradient and velocity can be visualized as a loop in phase space. The enclosed loop area is shown to equal the mean hydraulic power per cycle, establishing an exact geometric-energetic identity that holds for arbitrary Womersley number and harmonic composition. We further show that higher harmonic content and increasing Womersley number induce topological transitions in these loops, producing cusps, self-intersections, and curvature hot spots that correspond to inertial-viscous phase dispersion. This framework provides a rigorous baseline for interpreting arterial pressure-flow loops, connecting frequency-domain impedance analysis to measurable time-domain geometry.