Numerical Simulation of Two-Dimensional Non-Linear Elastic Membrane Waves under Radial Gradient Tension Field: A Case Study on Galaxy Ring Formation Topology

We present a self-contained finite-difference case study of two-dimensional non-linear elastic membrane wave propagation, motivated by the multi-ringmorphology of the ``Bullseye'' galaxy LEDA~1313424. The governing equation combines a tension-driven wave term, a Kirchhoff--Love bending-stiffness(biharmonic) dispersion term, and a state-dependent non-linear damping term coupled to an irreversible Continuum Damage Mechanics (CDM) variable. Alocalized Gaussian velocity impulse at the domain center is used to model a perpendicular perturber crossing a tensioned membrane. We first establish thatthis point-source, finite-speed wave architecture reproduces a clean bifurcation between a quasi-elastic, multi-ring regime (up to nine resolvablering peaks) at low impact velocities and a plastic-yield, shock-collapse regime (one to two ring peaks) at high impact velocities, controlled entirelyby a single tunable parameter, the impact velocity $v_0$. We then test whether replacing the uniform membrane tension with a radially decayingprofile $T(r) = T_0/\bigl(1+(r/R_{\rm core})^2\bigr)$, intended to emulate the stress concentration of a centrally peaked galactic mass distribution,spontaneously reproduces the observed inner-dense/outer-sparse ($\propto 1/n$-type) ring spacing reported for LEDA~1313424. Across a fullinteger sweep of impact velocities $v_0 = 1$ to $14$, and across a wide range of core radii $R_{\rm core}$, the simulated ring spacing instead remains\emph{monotonically decreasing} with radius (outer-dense/inner-sparse), the mirror opposite of the observational trend. We use ray-theoretic (WKB)arguments to show that this outcome is a structural consequence of finite-speed,causal wave propagation from a localized source, and is not resolved by anychoice of tension-gradient steepness. We explicitly distinguish this finding as a \emph{model-topology boundary limitation} of the continuum-membraneanalogy not as evidence for a new physical mechanism and we situateit against the local epicyclic-oscillator framework used in the CollisionalRing Galaxy (CRG) and dark-matter caustic-ring (CRDM) literature, which relies on a causally consistent tidal-impulse approximation rather than wavepropagation. All numerical data, source code, and figures referenced in this note are reproducible from the accompanying script.Keywords: non-linear wave equation, continuum damage mechanics, finite-difference time-domain,dispersion, WKB approximation, ring galaxies, numerical case study.