Multimodal nonlinear acoustics in two- and three-dimensional curved ducts

Overview

This work develops a weakly nonlinear model of duct acoustics extended to three dimensions without flow, building upon prior two-dimensional analysis. The model accommodates general curvature and width variation in two-dimensional ducts, and general curvature and torsion with radial width variation in three-dimensional ducts. The governing equations of gas dynamics are perturbed and expanded to second order to capture wave steepening and weak shock formation. The framework combines the curved-duct linear multi-modal method with straight-duct nonlinear multimodal techniques, resulting in an infinite set of coupled ordinary differential equations for modal coefficients. A linear matrix admittance and its weakly nonlinear generalization as a tensor convolution are solved throughout the duct and subsequently used to determine acoustic pressures and velocities. The admittance independently encodes the acoustic and weakly nonlinear properties of the duct geometry separate from the specific wave source, providing both computational tractability and physical insight. The model is validated against known configurations and explored through numerical examples demonstrating effects of torsion, curvature, width variation, and nonlinearity. Potential applications include sound propagation in brass instruments, where nonlinear wave steepening contributes to tonal characteristics and curvature affects effective duct length.

Methods and approach

The mathematical framework perturbs the equations of gas dynamics and expands them to second order to allow for wave steepening and weak shock formation. The resulting equations are expanded temporally in a Fourier series and spatially in terms of straight-duct modes. A multi-modal method is applied, projecting the curved-duct acoustics equations for pressure and velocity onto a basis of straight-duct modes. This projection converts the governing partial differential equations into an infinite coupled set of ordinary differential equations for the amplitude of each mode. The admittance, defined as the ratio of velocity to pressure, is introduced and a Riccati-style equation for the admittance is solved. A modified mathematical framework is developed that unifies both two- and three-dimensional cases, improving computational efficiency compared to previous two-dimensional analyses. The approach uses linear matrix and nonlinear tensor convolution notation to manage the algebraic complexity introduced by varying duct geometry. The framework is numerically implemented with truncation of the infinite mode set, and source code is provided as supplementary material.

Key Findings

Numerical examples validate the model against straight duct configurations, constant curvature bends in both two and three dimensions, and exponential horn geometries. The analysis demonstrates acoustic leakage from ducts owing to both curvature and nonlinearity. Results compare two- and three-dimensional behavior and quantify the effects of torsion, curvature, and width variation on acoustic propagation. The model reveals variation in the effective duct length of a curved duct as acoustic amplitude changes, addressing questions of pitch stability in musical instruments at different sound levels. The admittance formulation provides insight into how downstream duct geometry affects acoustic behavior independently of upstream wave source characteristics. The framework successfully captures the interplay between nonlinearity and duct geometry, including width variation, curvature, and torsion in three dimensions. The improved computational efficiency enables analysis of a range of curved geometries in both linear and nonlinear regimes that were previously prohibitive.

Implications

The model provides a framework for investigating nonlinear wave steepening in curved ducts, directly relevant to understanding tonal characteristics in brass instruments where curvature is geometrically necessary for practical design. The ability to analyze how nonlinear steepening affects effective duct length enables exploration of instrument design for pitch stability across varying sound amplitudes. The admittance formulation offers potential for understanding duct acoustic properties independent of excitation sources, which may inform design optimization. The extension to three dimensions with general curvature and torsion represents a substantial advancement beyond previous two-dimensional analyses, enabling realistic modeling of complex duct geometries found in actual instruments. The computational efficiency improvements make the framework tractable for practical analysis. Future applications may include detailed acoustic modeling of brass instruments where both geometric complexity and nonlinear effects are significant. The framework may also extend to other applications involving weakly nonlinear acoustics in complex geometries beyond musical instruments.