The Weakly-Nonlinear Admittance at Open Ends of Two- and Three-Dimensional Acoustic Waveguides

Abstract

We formulate a weakly-nonlinear exit condition for open ends of acoustic waveguides; to our knowledge this is the first time a general acoustic open end has been analysed outside the linear regime. The formulation neglects mean flow, vortex separation, and viscous effects. The resulting admittance boundary condition, and its weakly-nonlinear counterpart, extend recent weakly-nonlinear modelling of curved ducts(Jensen & Brambley 2026, arXiv:2503.11536) to include open ends. We approximate free space by considering the open-ended duct to be enclosed within a much larger hard-walled concentric duct; within the larger duct, the smaller duct exit is modelled as an acoustic discontinuity. Importantly, the superposition principle is unneeded, allowing the model to be applied in the nonlinear regime. The exit condition can be calculated without needing to solve the full problem in either the outer or inner ducts, making it numerically efficient. The exit condition is validated in the linear regime by comparison to Wiener-Hopf solutions of the duct end correction, and a novel nonlinear end correction is proposed; we find that both non-plane waves and nonlinearity cause the end correction to vary significantly from Rayleigh's classical 0.6 radii result. A number of numerical illustrations are then discussed, demonstrating nonlinear effects, sound radiating from curved ducts, sound radiating from an exponential horn (representative of brass instrument bells), and the harmonic effects of the open end on in-duct resonances. The model has potential applications to sound in woodwind and brass instruments. Matlab source code is provided in the supplementary material.