Nonlinear Proudman Resonance Under Moving Atmospheric System

A nonlinear theoretical model with an exact solution is developed to critically examine the Proudman resonance problem. In this framework, a moving atmospheric pressure system forces a wave governed by the square of the wave Froude number (n=F2). Proudman’s classical linear solution, although widely cited, diverges to infinity at resonance (n=1), limiting its physical realism. This work extends the linear regime into the nonlinear by solving the governing equations exactly. The resulting solution remains finite at resonance. The first-order approximation recovers Proudman’s solution, while the second-order approximation already yields a finite resonant response—providing a more accurate and physically realistic alternative. The analysis reveals that nonlinear advection significantly suppresses resonance amplification. At resonance, although the exact solution is finite, it bifurcates into two branches. Furthermore, while the linear Proudman's solution does not specify any limitations on the nature of the atmospheric pressure system, the new results show that resonant solutions exist for all low-pressure systems but are only conditionally valid for high-pressure systems: At low speeds, a high-pressure system can still produce a forced wave solution, but as the speed increases, the solution quickly becomes invalid. The closer n is to 1, the more likely the solution does not exist under a high-pressure system. Finally, the comparison with the exact solution under low-pressure systems reveals that Proudman’s linear approximation exhibits greater errors in both supercritical (n>1) and subcritical (n<1) regimes as the Froude number gets closer to 1.