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

TL;DR

The paper addresses how open-ended acoustic waveguides radiate in the weakly nonlinear regime without a mean flow, proposing an open-end boundary condition derived from a concentric outer duct that approximates free space. It combines mode-matching with a weakly nonlinear expansion to order $M^2$, yielding a Riccati-type system for admittances and a nonlinear generalization of the end-correction, which is validated against Wiener–Hopf solutions in the linear limit. Key contributions include (i) a physically grounded exit condition that works for arbitrary duct geometries, (ii) a generalized, nonlinear end-correction coefficient that depends on $M$, frequency $\omega$, and duct length $L$, and (iii) extensive numerical demonstrations across 2D and 3D geometries (straight, curved, and exponential-horn ducts) illustrating nonlinear radiation effects and resonance modifications. The approach enables efficient nonlinear analysis of duct acoustics and has potential applications to brass and woodwind instrument design, end-tuning, and the study of harmonic content in open-ended resonators.

Abstract

We formulate a weakly-nonlinear exit condition for open ends of acoustic waveguides without mean flow; to our knowledge this is the first time an acoustic open end has been analysed outside the linear regime. The resulting admittance boundary condition, and its weakly-nonlinear counterpart, extends recent weakly-nonlinear modelling of curved ducts (Jensen & Brambley 2025, arXiv:2503.11536) to include open ends. We approximate free space by considering the open-ended duct to be enclosed within a much larger 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.

Figures (14)

• Figure 1: A schematic depicting the inner and outer ducts. (a) The inner duct, with centreline $\mathbf{q}(s)$, inlet ($s = 0$), and the outlet boundary condition/surface of emission ($s = s_\text{o}$) depicted in green. (b) The outer duct, with centreline $\mathbf{q}(s)$, relative duct widths $X_1$ and $X_2$, the acoustic source/surface of emission ($s = 0+$) depicted in green, the absorbing boundary or 'acoustic wormhole' ($s=0-$) depicted in red, and the two domains of propagation $s > 0$ and $s < 0$, with acoustic variables continuous across the dash-dotted line. Note that the outer duct domain $s<0$ intersects with the inner duct domain, shown by the blue dashed lines, but this intersection is ignored in this model. Outer and inner widths $X_2$ and $X_1$ use the 2D notation.
• Figure 2: End-correction coefficients for a 3D duct, calculated using the Wiener--Hopf technique (solid line) and our outlet condition (dash-dotted line), for $m_\alpha=0$; 8 modes were used in the inner duct and 1200 in the outer duct, with a duct width ratio of 1/40.
• Figure 3: End-correction coefficients for a 2D duct with a symmetric source, calculated using the Wiener--Hopf technique (solid line) and our outlet condition (dash-dotted line). 13 modes were used in the inner duct and 2000 in the outer duct, with a width ratio of 1/40.
• Figure 4: Comparison of different numerical parameter choices for a segment of figure \ref{['2Dendcorrectionplot.pdf']} between $2\pi$ and $4\pi$, with only the $\alpha^1 = 2$ mode shown. The Wiener--Hopf solution is plotted in black, and the original parameter choice $(\eta,\alpha^2_\mathrm{max}) = (1/40,2000)$ from figure \ref{['2Dendcorrectionplot.pdf']} is shown plotted with a green dashed line as before. All of these take $\alpha_\mathrm{max}^1 = 10$.
• Figure 5: The nonlinear generalised end correction coefficient $\tilde{s}_0/R_1$ for fixed Helmholtz number $\omega R_1 = 0.05$ and a range of duct lengths (scaled by source wavelength $\lambda = 2\pi/\omega$) and Mach numbers, in 3D. Nonlinearity is shown to have a dampening effect on the discrepancy between duct length and resonant length for low frequencies, and the longer the duct the stronger this effect will be. 8 modes were used in the inner duct and 200 in the outer duct, with 10 temporal modes and a width ratio of 1/10.