Dynamic Pressure Response and Wave Resistance in Forced Korteweg–DeVries Systems

Abstract

Weakly nonlinear free-surface flows past disturbances are traditionally modeled using the forced Korteweg–de Vries (fKdV) equation with a prescribed instantaneous pressure field. However, physical wake responses possess finite relaxation times and advection scales that diagnostic algebraic closures fail to capture. This paper introduces a novel coupled system in which the surface pressure is a dynamical field governed by an advection–reaction–diffusion equation driven by band-limited curvature. Using linear spectral theory and numerical validation, we derive a phase-speed criterion demonstrating that energy transfer is determined by the comparison between the pressure drift speed and the surface phase speed. A sharp stability theorem proves that, to leading order in the coupling strength and for a non-negative even response transfer function whose drift speed exceeds the Froude detuning, the system is spectrally stable if and only if the response is band-limited below a critical wavenumber kc. Furthermore, an exact energy identity establishes that passivity and linear stability are equivalent. Finally, we demonstrate resonance steering: while coupling typically increases the wave resistance for monotone spectra, tuning the response to a spectral zero of a multi-lobe footprint reduces the drag significantly relative to its classical value. This result identifies an explicit performance–strongness trade-off, providing a mathematically strong structure for wave drag minimization through dynamic pressure control.