Scaling Laws of Tidal Wave Dispersion: Multi-frequency, Multi-estuary Analysis

Tidal transmission in estuaries reflects a balance between amplitude damping and phase progression. Yet observations consistently show frequency-dependent variability that exceeds the explanatory reach of a single wave--diffusion model. To address this gap, we analyze boundary--inland gauge pairs from five U.S. estuaries and externally validate the findings in nine non-U.S. systems---Tokyo Bay (Japan), Port Phillip Bay (Australia), the Waitemat\={a} Harbour (Auckland, New Zealand), the Fraser River estuary (Canada), the Tamsui River estuary (Taiwan), and four European estuaries: Elbe and Weser (Germany) and Gironde and Loire (France). From these records, we quantify amplitude decay and phase lag for leading diurnal and semidiurnal constituents. The results reveal a striking asymmetry: phase accumulates roughly an order of magnitude faster than amplitude decays, placing most systems in a propagation-dominated, weak-memory regime. Instances of inland amplification occur but remain rare and do not dominate the population. To interpret this behavior, we propose a lagging framework with two characteristic timescales. From a minimal parameter set, the framework predicts frequency-dependent slopes, reproduces phase-lag behavior with near one-to-one accuracy, and captures damping with moderate scatter. When applied across systems, the results collapse onto two nondimensional controls: the ratio of forcing period to travel time and the ratio of storage to propagation timescales. We further synthesize constituent-wise slopes into estuary-level ''memory'' features and apply an unsupervised clustering, which objectively groups estuaries into low- to mid--low-memory archetypes---convergent or inlet-restricted, tidal--fluvial, and a non-passive reflection class---consistent with independent geomorphic descriptors. All nine non-U.S. estuaries fall within the same low-memory envelopes, underscoring the framework's generality. A companion distance--frequency chart translates these fits into estimates of tidal penetration, revealing shorter inland reaches for semidiurnal than for diurnal tides. Overall, the analysis offers a compact mechanistic view of estuarine memory and a demonstrably transferable tool for network design, forecasting, and change detection.