Webster Horn Equation Research Articles

Fast design of non-uniform acoustic waveguides with prescribed eigenfrequencies

In this article we address the issue of finding acoustic and elastic resonator shapes corresponding to given eigenfrequencies. This problem has a wide range of applications spanning from ultrasonics through the music acoustics to metamaterials. We employ the previously published solution to the Webster horn equation and use it to train an artificial neural network to solve the inverse eigenvalue problem. Subsequently, it is shown how the isospectrality and scalability properties might provide a whole class of fast designed resonator shapes.

Using a one-dimensional finite-element approximation of Webster's horn equation to estimate individual ear canal acoustic transfer from input impedances

Knowledge of the sound pressure transfer to the eardrum is important. The transfer is highly influenced by the shape of the ear canal and its acoustic properties, such as the acoustic impedance at the eardrum. Invasive procedures to measure the sound pressure at the eardrum are usually elaborate or costly. We propose a numerical method to estimate the transfer impedance of the ear canal given only input impedance measurements at the ear canal entrance, by using one-dimensional first-order finite elements and Nelder-Mead optimization algorithm. Estimations on the area function of the ear canal and the acoustic impedance at the eardrum are achieved. Results are validated through numerical simulations on ten different ear canal geometries and three different acoustic impedances at the eardrum, using synthetically generated data from three-dimensional finite element simulations.

Analytical Solutions for Side-branch Impedance Producing Spatial Localization of Acoustic Waves in Ducts with Varying Cross Sections

Analytical mathematical models and solutions for spatial localization of acoustic waves through an impedance discontinuity produced by an intermediate damped side branch are studied in stationary media in ducts with varying cross sections. Three specific geometries, namely, with polynomial, sinusoidal, and exponential longitudinal variations, are investigated. The sound fields inside the ducts are modeled by Webster's horn equation. Traveling-wave solutions are obtained by appropriate transformations. The side-branch impedances required for spatial localization (confinement) of traveling and standing waves are found analytically and verified numerically using three-dimensional finite element analysis. The impact of the longitudinal variation of the duct's cross-sectional area (CSA) on the side-branch impedance is examined. It was found that the required side-branch resistance changes more than the reactance with the variation of the duct CSA. A measure of a traveling wave is defined to quantitatively examine the spatial localization of acoustic waves. It was found that the CSA corrections on the side-branch impedances are important. The results of this study reveal the quantitative relationships between the side-branch impedance and the CSA variations for zero reflection from the impedance discontinuity. The mathematical approach presented is potentially helpful for a design of a full anechoic termination and energy localization in duct systems.

High frequency propagation in a waveguide with an expanding section

A waveguide whose walls have a constant local impedance is readily analyzed if its cross section is uniform with a standard shape. This is not true if the cross section expands or contracts. The (one-dimensional) Webster horn equation typically is used for low frequency propagation in such configurations, but doing so restricts consideration to situations where the size of the cross section changes slowly over the scale of a wavelength. Somewhat paradoxically, a waveguide consisting of a sequence of different sized uniform segments has been the subject of numerous analyses that are not limited to low frequencies. The present work extends one such analysis, which uses a collocation technique at discontinuous junctions, to a two-dimensional configuration in which a transition section expands linearly. The field in each section is described as a modal series, whose terms are the product of a transverse standing mode and two phasors for axial propagation. Cartesian coordinates are used to describe the field within the uniform sections, whereas a polar coordinate description of the field in the expansion section facilitates satisfying rigid wall conditions. Various schemes for arranging the collocation points are discussed, and convergence of the series is examined.

Design of radial sonic crystal for sound attenuation from divergent sound source

Sonic crystals are periodic arrangement of sound hard scatterers, generally in square or triangular lattice configuration. The periodic obstruction to the sound wave by the scatterers leads to an interesting phenomenon of the band gap, which results in a high sound attenuation in the band gap region. In this work, a design of sonic crystal called as the radial sonic crystal is presented, which consists of periodic structures in polar coordinates. Such a structure attenuates divergent sound source. The radial sonic crystal is designed based on the Webster horn equation and using the property of invariance of governing equation from one unit cell to another. The designed radial sonic crystal is tested experimentally and by the finite element simulation. The experimental results are in good agreement with the simulation and show high sound attenuation of 30 dB. The high sound attenuation is due to the presence of the band gap in the radial sonic crystal.

Acoustic horn reflectance: Equations and measurements

Reflectance is the transfer function between forward and reflected components of pressure waves that propagate in wave guides such as acoustic horns. Exact solutions to Webster's Horn Equation are only known for a few specific shapes, including parabolic, conical, and exponential. Explicit equations for reflectance in these three horn shapes were recently published for infinite-length horns. Measured reflectance in 3D-printed, finite-length examples of these horn shapes, show no similarity in the frequency-domain to exact reflectance for infinite-length horns. The similarity improves after adjustments to both the equations and the measurements. New equations were derived for exact reflectance of finite-length horns. Measured reflectance was smoothed by time-domain windowing. In contrast to frequency-domain reflectance, comparisons of time-domain reflectance prior to the time sound reaches the end of the horn were not much affected by these adjustments. Because exact equations are known and 3D-printed examples are easy to obtain, these three horn shapes may be useful as standards for comparing different reflectance-measurement systems.

Spectrally accurate numerical solution of acoustic wave equations

Finite difference models of wave propagation have presented challenging problems of stability and accuracy since initial experimentation with these models began on early digital computers. The advent of spectral methods in the late 1960s has led to the latter's increasing use for solving differential equations in a range of fluid dynamic, electromagnetic and thermal applications. Spectral methods transform a physical grid of state variables (such as acoustic velocity and pressure) into an alternative spectral space characterized by a particular set of basis functions. Spatial derivatives of physical state variables are computed in spectral space using exact differential operators expressed in terms of those functions. Fast numerical transforms are employed to exchange immediate state of a simulation between its spectral and physical representations. In problems equally suited to spectral and finite difference formulation, spectral methods often yield increased fidelity of physical results and improved stability. Spectral methods sometimes enable computational grid size requirements of a simulation to be substantially reduced, with concomitant computational savings. This paper reports on spectral implementations of the acoustic wave equation and Webster horn equation for simulating audio transducer cavities, musical instrument resonators, and the human vocal tract.

A class of damping models preserving eigenspaces for linear conservative port-Hamiltonian systems

For conservative mechanical systems, the so-called Caughey series are known to define the class of damping matrices that preserve eigenspaces. In particular, for finite-dimensional systems, these matrices prove to be a polynomial of one reduced matrix, which depends on the mass and stiffness matrices. Damping is ensured whatever the eigenvalues of the conservative problem if and only if the polynomial is positive for positive scalar values.This paper first recasts this result in the port-Hamiltonian framework by introducing a port variable corresponding to internal energy dissipation (resistive element). Moreover, this formalism naturally allows to cope with systems including gyroscopic effects (gyrators).Second, generalizations to the infinite-dimensional case are considered. They consist of extending the previous polynomial class to rational functions and more general functions of operators (instead of matrices), once the appropriate functional framework has been defined. In this case, the resistive element is modelled by a given static operator, such as an elliptic PDE. These results are illustrated on several PDE examples: the Webster horn equation, the Bernoulli beam equation; the damping models under consideration are fluid, structural, rational and generalized fractional Laplacian or bi-Laplacian.

Sound propagation in two-dimensional waveguide with circular wavefront

Sound propagation through a waveguide is generally modeled by the Webster horn equation which assumes a planar pressure wavefront. However, most of the sources are non-planar in nature. In this work, a 1-D model is derived for sound propagation through a 2-D waveguide with circular wavefront. The model is derived from the 2-D Helmholtz equation using the weighted residual method. The model assumes a uniform pressure across the angular coordinate at a given radial distance. A 2-D finite element model is used to validate the results for different waveguide geometries and it shows good agreement.

A quasi two-dimensional model for sound attenuation by the sonic crystals

Sound propagation in the sonic crystal (SC) along the symmetry direction is modeled by sound propagation through a variable cross-sectional area waveguide. A one-dimensional (1D) model based on the Webster horn equation is used to obtain sound attenuation through the SC. This model is compared with two-dimensional (2D) finite element simulation and experiment. The 1D model prediction of frequency band for sound attenuation is found to be shifted by around 500 Hz with respect to the finite element simulation. The reason for this shift is due to the assumption involved in the 1D model. A quasi 2D model is developed for sound propagation through the waveguide. Sound pressure profiles from the quasi 2D model are compared with the finite element simulation and the 1D model. The result shows significant improvement over the 1D model and is in good agreement with the 2D finite element simulation. Finally, sound attenuation through the SC is computed based on the quasi 2D model and is found to be in good agreement with the finite element simulation. The quasi 2D model provides an improved method to calculate sound attenuation through the SC.

Reflectance of acoustic horns and solution of the inverse problem.

A method is described for solving the inverse problem of determining the profile of an acoustic horn when time-domain reflectance (TDR) is known only at the entrance. The method involves recasting Webster's horn equation in terms of forward and backward propagating wave variables. An essential feature of this method is a requirement that the backward propagating wave be continuous at the wave-front at all locations beyond the entrance. Derivation of the inverse solution raises questions about the meaning of causality in the context of wave propagation in non-uniform tubes. Exact reflectance expressions are presented for infinite exponential, conical and parabolic horns based on exact solutions of the horn equation. Diameter functions obtained with the inverse solution are a good match to all three horn profiles.

On damping models preserving the eigenfunctions of conservative systems: a port-Hamiltonian perspective

In this paper, a special class of damping model is introduced for second order dynamical systems. This class is built so as to leave the eigenfunctions invariant, while modifying the dynamics: for mechanical systems, well-known examples are the standard fluid and structural dampings.In the finite-dimensional case, the so-called Caughey series are a general extension of these standard damping models; the damping matrix can be expressed as a polynomial of a matrix, which depends on the mass and stiffness matrices. Damping is ensured whatever the eigenvalues of the conservative problem if and only if the polynomial is positive for positive scalar values. This can be recast in the port-Hamiltonian framework by introducing a port variable corresponding to internal energy dissipation (resistive element). Moreover, this formalism naturally allows to cope with systems including gyroscopic effects (gyrators).In the infinite-dimensional case, the previous polynomial class can be extended to rational functions and more general functions of operators (instead of matrices), once the appropriate functional framework has been defined. In this case, the resistive element is modelled by a given static operator, such as an elliptic PDE. These results are illustrated on several PDE examples: the Webster horn equation, the Bernoulli beam equation; the damping models under consideration are fluid, structural, rational and generalized fractional Laplacian or bi-Laplacian.

Analysis of frequency band structure in one-dimensional sonic crystal using Webster horn equation

Sound propagation through periodic arrangement of scatterers lead to formation of bands of frequencies, known as band gaps, where sound cannot propagate though the structure. We propose a method based on Webster horn equation, along with Floquet theorem, to predict the band gap of a one-dimensional periodic structure made of hard sound-scatterers. The method is further modified to obtain the complex wave numbers, which give the decay constants. The decay constant is used to predict the sound attenuation of the evanescent wave in the finite sonic crystal. The theoretical prediction is verified with experimental measurements.

Optimal Control Theory: A Method for the Design of Wind Instruments

It has been asserted previously by the author that optimal control theory can be a valuable framework for theoretical studies about the shape that a wind instrument should have in order to satisfy some optimization criterion, inside a fairly general class. The purpose of the present work is to develop this new approach with a look at a specific criterion to be optimized. In this setting, the Webster horn equation is regarded as a controlled dynamical equation in the space variable. Pressure is the state, the control being made of two parts~: one variable part, the inside diameter of the duct and one constant part, the weights of the elementary time-harmonic components of the velocity potential. Then one looks for a control that optimizes a criterion related to the definition of an {oscillation regime} as the cooperation of several natural modes of vibration with the excitation, the {playing frequency} being the one that maximizes the total generation of energy, as exposed by A.H. Benade, following H. Bouasse. At the same time the relevance of this criterion is questionned with the simulation results.

Nonlinear frequency shifts in acoustical resonators with varying cross sections

The frequency response and nonlinear resonance frequency shift of an acoustical resonator with losses and having a varying cross section were investigated previously using Lagrangian mechanics and perturbation for resonator shapes that are close to cylindrical [M. F. Hamilton, et al., J. Acoust. Soc. Am. 110, 109-119 (2001)]. The same approach is extended here to include resonators having any shape for which the Webster horn equation is a valid model in the linear approximation. Admissible shapes include cones and bulbs proposed for acoustical compressors. The approach is appropriate for approximate but rapid parameter estimations for resonators with complicated shapes, requiring far less computation time than for direct numerical solution of the one-dimensional model equation frequently used for such resonators [Ilinskii et al., J. Acoust. Soc. Am. 104, 2664-2674 (1998)]. Results for cone and bulb shaped resonators with losses are compared with results from the direct numerical solution. The direction of the resonance frequency shift is determined by the efficiency of second-harmonic generation in modes having natural frequencies below versus above the frequency of the second harmonic, and how the net effect of this coupling compares with the frequency shifts due to cubic nonlinearity and static deformation.

Acoustic tubes with maximal and minimal resonance frequencies

This paper presents a theoretical derivation of acoustic tract shapes that minimize and maximize resonance frequencies. The derivation is based on a symmetry of Webster's horn equation and on Ehrenfest's adiabatic invariance hypothesis. It is shown that for minimizing formant frequencies, abrupt transitions are necessary, while for maximizing resonance frequencies, gradual transitions are needed. It is argued that this has implications for modeling human, animal and prehistoric vocal tracts. Such models should represent the anatomic (in)ability to produce abrupt and gradual transitions correctly, otherwise they would have biases towards different sets of formant frequencies than real vocal tracts.

The spatial distribution of sound pressure within the human ear canal

The sound field inside a human ear canal has been computed using 2 approaches. A simple model, a modified Webster horn equation approach, can accommodate the curvature and varying cross section of the ear canal. Calculations using the horn equation demonstrate the formation of standing waves within the canal. The pressure may be interpreted as either the pressure along the center axis of the canal or the average pressure within a cross-sectional slab. To investigate possible spatial variation through a cross section, the sound field has also been computed using the boundary element method (BEM). Over 2000 triangular mesh elements, 1 mm or less in size, were used to represent the canal geometry. For a plane piston source at the canal entrance and both a rigid and a resistive impedance condition at the eardrum position, the computed sound pressures along the center axis of the ear canal are in good agreement with the horn equation calculations, up to 15 kHz. The BEM approach, though, reveals spatial variations of sound pressure through each canal cross section, increasingly significant as frequency increases. Further, for source configurations that are more realistic than a simple piston, large transverse variations in sound pressure are anticipated.

Webster's Horn Equation Revisited

The problem of low-frequency sound propagation in slowly varying ducts is systematically analyzed as a perturbation problem of slow variation. Webster's horn equation and variants in bent ducts, in ducts with nonuniform soundspeed, and in ducts with irrotational mean flow, with and without lining, are derived, and the entrance/exit plane boundary layer is given. It is shown why a varying lined duct in general does not have an (acoustic) solution.

Acoustic waves in slender axisymmetric tubes

Acoustic waves in a rigid axisymmetric tube with a variable cross-section are considered. The governing Helmholtz equation is solved using Neumann series (expansions in Bessel functions of various orders) with a stretched radial coordinate, leading to a hierarchy of one-dimensional ordinary differential equations in the longitudinal direction. The lowest approximation for axisymmetric motion includes Webster's horn equation as a special case. Fourth-order differential equations are obtained at the next level of approximation. Good agreement with existing asymptotic theories for waves in slender tubes is found.

AN ACOUSTIC TRANSFORMER COMPOSED OF HORNS AND ITS APPLICATION

A new acoustic transformer was developed by connecting three horns to improve radiation performance in the frequency region below 500 Hz. The proposed acoustic transformer was evaluated by numerical analysis using the commercial computer program SYSNOISE and by experiment. The acoustic transformer is composed of three horns, one of which was used in an inverted form. A design model was developed by use of Webster's horn equation and showed that the transformer can improve radiation efficiency. This was confirmed by numerical calculation using SYSNOISE. An acoustic projector was designed by use of the developed transformer and a piezoelectric unimorph-type actuator. The sound pressure measured at the mouth of the constructed acoustic projector was compared with the sound pressures evaluated at the same location by numerical calculation to investigate the differences between the numerical simulation model and the actual acoustic projector. Sound pressures generated by several acoustic radiators were calculated numerically and compared with the measured and calculated sound pressures of the developed acoustic projector to evaluate the effects of use of the proposed acoustic transformer. The comparative evaluation shows that the proposed acoustic transformer can provide up to a 10 dB gain over use of a horn in the narrow band low-frequency region from 100 to 200 Hz.