Acoustic wave propagation in hard-walled ducts is of interest in many fields including vehicle design, musical instruments acoustics, and architectural and environmental noise-control. For the case of small sinusoidal perturbation of the cross-section, it is possible to derive simple though approximate analytical formulas of its plane wave acoustic reflection and transmission spectral response that resembles the optical situation of uniform Bragg gratings. The proof is given here, starting from the “horn equation” and then exploiting the coupled-modes theory. Examples of the results obtained with these analytical formulas are shown for some sinusoidally perturbed ducts and compared to results obtained through a numerical method, revealing a very good agreement.

The propagation of waves in periodic media has received much attention in the past in different fields of physics: a comprehensive review can be found in Elachi [

In the last decades much research effort was dedicated to the theoretical and experimental study of elastic wave propagation in periodic waveguides. For instance, Fokkema [

The purpose of this paper is not to advance the research work accomplished so far, but rather to provide an approximate simplification of the established theory, when proper hypotheses are satisfied. Attention is indeed limited to acoustic propagation in hard-walled ducts whose cross-section undergoes a small sinusoidal perturbation with respect to a reference mean value. The work holds for any filling fluid, typically air, provided that the hard-wall hypothesis is verified.

Like in Munday et al. [

Recently, Hawwa [

In optics, or more general in electromagnetics, a periodically perturbed medium is called a Bragg grating, or simply a multilayer medium: the transmission and reflection of a uniform grating can be expressed with simple closed-form formulas (Kogelnik [

This paper exploits thus the Bragg gratings theory to solve the acoustic “horn equation,” obtaining simple formulas for the reflection and transmission spectral response of the waveguide as a function of acoustic and geometric parameters, under the hypothesis of a small sinusoidal cross-section variation compared to the mean reference value.

Simple and closed-form solutions are advantageous in modeling/inversion procedures, as, for example, in bore reconstruction, design of noise-control devices (employed, e.g., for jet engines or HVAC systems), and even monitoring of transportation pipelines.

The following sections describe the scenario, the mathematical derivation, and the result for some example cases.

The solution provided can take into account also wave attenuation, typically hard-wall losses (boundary layer friction), whose terms are added a posteriori.

The scenario considered here is a circular hard-walled duct with a small periodic perturbation of the cross-section along the axial coordinate (Figure

Duct with a periodic perturbation of the cross-section.

As in Lau and Campos [

Under these hypotheses the governing geometrical parameters reduce to the cross-section area (the shape can be neglected), and the starting physical law is the one-dimensional “horn equation” [

Let us consider harmonic waves:

The substitution of (

From this point, the mathematical steps and the approximations introduced follow the computation of the transfer function for a uniform optical fiber Bragg grating (Erdogan [

The pressure field is expressed as a linear combination of the fundamental modes propagating in the opposite directions

The first derivative of (

In the hypothesis of weak coupling between the two modes, the second derivatives of

Substituting the expressions for derivatives, and multiplying by

The resulting system is

The ratio between the derivative of the cross-section and the cross-section itself

The system (

The result is

Since

The solution of system (

The reflection and transmission spectral responses

According to their definition, the spectral responses

Equations (

The reflection spectral response has a maximum for

Finally, even if the considered one-dimensional wave equation does not contain a loss term, one may add the absorption phenomenon a posteriori, typically due to the duct walls, by redefining the acoustic wavenumber

Some examples of the acoustic behavior for a sinusoidally perturbed duct, filled with air at standard conditions (20°C, 1 atm), are presented here. They reveal the link between the passbands/stopbands response and the waveguide geometrical parameters, and they permit inferring the validity limit of the approximate formula.

Wave attenuation and dispersion are included too, using (

We consider a sinusoidal duct with length

The other parameters are set to the following values:

Case (i):

Case (ii):

Case (iii):

Figure

Reflection ((a), (c), (e)) and transmission ((b), (d), (f)) spectral responses, in magnitude, of a sinusoidally perturbed duct computed with (

The crosses in Figure

The parameters values have been chosen to progressively increase the

The difference between approximate and theoretical values is barely noticeable even when

The phases are not shown here, but they have been verified as well.

The acoustic spectral response of a sinusoidally perturbed hard-wall duct has been derived and given in a simple formula, by following the optical analog of Bragg gratings. The formula is the same as in optics, with the electromagnetic parameters replaced by their equivalent acoustic parameters and periodic duct geometry.

Results are valid for small cross-section perturbations and in this case successful comparisons with a numerical method are shown, even in case of wave attenuation.

The availability of simple analytical formulas permits a direct analysis of the link between the acoustic response and the duct geometrical parameters and the design of efficient modeling/inversion procedures in the fields of bore/pipe reconstruction, noise control, and so forth.

Finally it can be noticed that, even if the derivation strictly requires a sinusoidal perturbation, any cross-section deformation can be decomposed in sinusoidal functions and therefore the results can be applied, provided that the underlying hypotheses are satisfied, to a much broader range of scenarios.

The authors wish to thank professor Andrea Melloni for the profitable discussions on the Bragg gratings theory and for the comments and suggestions on the paper production.