Longitudinal versus quasi-P waves in weakly anisotropic elasticity: operator comparison and long-time dephasing

70/2026

Longitudinal versus quasi-P waves in weakly anisotropic elasticity: operator comparison and long-time dephasing

Thursday 24th September 2026

Micheletti, S.

In isotropic elasticity, P-wave displacement is longitudinal and curl-free. In anisotropic elasticity, the physical quasi-P (qP) mode is instead selected by the principal eigenvector of the Christoffel matrix and is generally not parallel to the wave normal. We compare, in homogeneous two-dimensional vertical transverse isotropy (VTI), the exact elastic qP propagator with a longitudinally constrained propagator that suppresses transverse displacement by construction. Both dynamics are placed in a periodic operator and variational framework. The qP space is an invariant spectral subspace of the elastic operator, whereas the longitudinal space yields a Galerkin restriction that is generally not invariant. The corresponding forms are coercive on mean-zero $H^1$, their generators are positive self-adjoint operators of order two, and both Cauchy problems are well posed and energy conservative. A two-dimensional Rayleigh-quotient identity expresses the longitudinal squared-speed deficit as the qP--qSV spectral gap multiplied by the squared polarization mismatch. Building on classical weak-anisotropy perturbation theory, we derive an explicit VTI specialization of the qP projector and phase symbol. The projector error and qSV contamination are $O(\eta)$ in amplitude, whereas the qP phase-speed error is $O(\eta^2)$. For longitudinal initial data, the field error admits an orthogonal decomposition into modal contamination and qP dephasing. A uniform estimate bounds it by $C|\eta|\|u_0\|+Ct\eta^2\|u_0\|_{H^1}$ on bounded times. After removal of the common carrier, an abstract two-branch multiplier theorem gives field-level limits on the intermediate scale $t=\tau/\eta$ and the diffractive scale $t=\tau/\eta^2$. The first yields a finite polarization--phase crossover, whereas the second yields a generally order-one dephasing profile. Exact-in-time Fourier experiments verify the identities, asymptotic coefficients, convergence rates, and two long-time limits. The main contribution is therefore not a new local qP approximation, but a quantitative error theory describing when a longitudinal elastic model approximates the exact qP branch and when its second-order phase accuracy is exhausted by long-time dephasing.

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Wave Motion (Elsevier) — accepted for publication, manuscript no. WAMOT-D-26-00490R1.