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-ring morphology 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. A localized Gaussian velocity impulse at the domain center is used to model a perpendicular perturber crossing a tensioned membrane. We first establish that this point-source, finite-speed wave architecturereproduces a clean bifurcation between a quasi-elastic, multi-ring regime (up to nine resolvable ring peaks) at low impact velocities and a plastic-yield, shock-collapse regime (one to two ring peaks) at high impact velocities, controlled entirely by a single tunable parameter, the impact velocity v0. We then test whether replacing the uniform membrane tension with a radially decaying profile T (r) = T0/1 + (r/Rcore)2, intended to emulate the stress concentration of a centrally peaked galactic mass distribution, spontaneously reproduces the observed inner-dense/outer-sparse (∝ 1/n-type) ring spacing reported for LEDA 1313424. Across a full integersweep of impact velocities v0 = 1 to 14, and across a wide range of core radii Rcore, the simulated ring spacing instead remains monotonically decreasing with radius (outer-dense/inner-sparse), the mirror opposite of the observational trend. We use ray-theoretic (WKB) arguments to showthat this outcome is a structural consequence of finite-speed, causal wave propagation from a localized source, and is not resolved by any choice of tension-gradient steepness. We explicitly distinguish this finding as a model-topology boundary limitation of the continuum-membrane analogy not as evidence for a new physical mechanism and we situate it against the local epicyclic-oscillator framework used in the Collisional Ring Galaxy (CRG) and dark-matter caustic-ring(CRDM) literature, which relies on a causally consistent tidal-impulse approximation rather than wave propagation. 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.