Longitudinal versus quasi-P waves in weakly anisotropic VTI media: a 2D analysis of modal error and long-time dephasing

In isotropic elasticity, P-wave displacement is longitudinal, whereas in anisotropic elasticity the quasi-P (qP) polarization is generally oblique to the wave normal. We quantify the resulting difference between exact elastic qP propagation and a longitudinally constrained model for homogeneous VTI media in the symmetry plane. The qP dynamics is the invariant spectral restriction of the full elastic operator; the longitudinal dynamics is a conservative Galerkin restriction to curl-free fields. An exact identity shows that the longitudinal squared-speed deficit equals the qP–qSV spectral gap times the squared polarization mismatch. Under weak anisotropy η , the modal/polarization error is O ( η ), while the phase-speed error is O ( η 2 ). Their propagated-field contributions are orthogonal and satisfy a bounded-time estimate O ( | η | + t η 2 ) . Strong L 2 limits on t = O ( η − 1 ) and t = O ( η − 2 ) identify, respectively, the polarization–phase crossover and order-one long-time dephasing. Exact Fourier experiments verify the asymptotic coefficients and limiting curves. The results provide a mechanical validity criterion for longitudinal P-wave models: short-time accuracy is limited by polarization leakage, whereas sufficiently long propagation is controlled by accumulated phase error.