Mathematical Problems in Engineering

Volume 2015, Article ID 516982, 8 pages

http://dx.doi.org/10.1155/2015/516982

## Sound Propagation in a Duct with Wall Corrugations Having Square-Wave Profiles

Department of Mechanical Engineering, King Fahd University of Petroleum and Minerals, Dhahran 31261, Saudi Arabia

Received 14 April 2015; Revised 17 June 2015; Accepted 18 June 2015

Academic Editor: Hakim Naceur

Copyright © 2015 Muhammad A. Hawwa. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Acoustic wave propagation in ducts with rigid walls having square-wave wall corrugations is considered in the context of a perturbation formulation. Using the ratio of wall corrugation amplitude to the mean duct half width, a small parameter is defined and a two levels of approximations are obtained. The first-order solution produces an analytical description of the pressure field inside the duct. The second-order solution yields an analytical estimate of the phase speed of waves transmitting through the duct. The effect of wall corrugation density on acoustic impedance and wave speeds is highlighted. The analysis reveals that waves propagating in a duct with square-wave wall corrugation are slower than waves propagating in a duct with sinusoidal wave corrugation having the same corrugation wavelength.

#### 1. Introduction

Acoustic waveguides with walls having weak periodic undulations have been modules of interest over the past decades. Although wall undulations constitute small features at the boundaries of the wave propagation domain, they can significantly transform acoustic wave propagation characteristics. Wall periodicities are commonly added patterned (periodic) corrugations that increase structural strength while keeping duct flexibility. In another scenario situation, geometric roughness on a duct wall can be looked at as irregular corrugations. Regardless, whether wall corrugations are deterministically periodic or statistically having a mean periodic profile, the dispersion spectrum of periodic acoustic waveguide is divided into passbands and stopbands.

As first steps towards dealing with the case of a waveguide with statistically rough boundaries, Samuels [1] and then Salant [2] employed a straightforward perturbation expansion for case of sound propagation in a two-dimensional (parallel plate) waveguide with small sinusoidal perturbations in the boundaries to study forward- and backward-propagating mode coupling. Nayfeh [3] employed a multiple scales perturbation technique to show that the perturbation expansion employed by Samuels [1] and Salant [2] was not uniform under resonance conditions and provided a uniformly valid perturbation expansion. Nusayr [4] considered the mode coupling case in a rectangular waveguide with sinusoidal boundaries, employing a multiple scales perturbation technique in the analysis. Boström [5] utilized a null field approach to study sound propagation in a cylindrical duct whose radius varied sinusoidally along its axial dimension and proved the existence of small passbands for the case of large-amplitude undulations. Asfar and Nayfeh [6] provided a review on using the perturbation method of multiple scales for wave propagation in periodic structures, including closed- and open-acoustic waveguides. Lundqvist and Boström [7] calculated transmission and reflection coefficients at the junction between a straight and a sinusoidally corrugated cylindrical duct using mode-matching under the context of a null field approach.

After 1990, Bradley [8] found Bloch wave functions to describe waves in a rectangular duct having rectangular side branches placed periodically and presented strong dispersion and sharply banded attenuation that could be used in a band reject travelling wave filter. Hawwa [9] presented a multiple scales perturbation approach for analyzing waveguides with surfaces having two periodicities and detailed scenarios of various incidents and reflected mode coupling in these waveguides. Hawwa [10] examined the influence of weighted tapered periodic undulations on acoustic wave reflection spectra in waveguides using perturbation analysis. Hawwa [11] investigated the effect of chirped periodic surface undulations on widening stopbands of acoustic wave reflection spectra in ducts by solving coupled mode equations describing modal interactions. Tao et al. [12] made a predictive study of non-Bragg resonance, caused by the interaction between the transverse standing acoustic waves occurring in a cylindrical waveguide with a sinusoidally perturbed wall. Potel and Bruneau [13] used standard integral Green’s formulation to describe the nonlocal mode coupling in an inviscid fluid-filled waveguide enclosed between two planes having two-dimensional irregular corrugations. Valier-Brasier et al. [14] considered acoustic eigenmode coupling in a fluid-filled waveguide enclosed between two planes having one-dimensional irregular corrugations having the shape of tiny parallel ridges. Tao et al. [15] showed the possibility of acoustic energy transmission within a Bragg stopband by means of high-order transverse modes interactions in an acoustic duct with periodically corrugated walls. Del Giudice and Bernasconi [16] utilized an acoustic horn equation to get reflection and transmission spectra in a waveguide with small sinusoidal cross section variation. Tao and Fan [17] used a perturbation method and a finite element formulation to consider transmission under non-Bragg resonances of three transverse modes in a waveguide with sinusoidally perturbed walls.

Almost all of the works cited above are concerned with sound propagating in waveguides with periodic wall corrugations and concentrate on wave modal interactions under Bragg and non-Bragg conditions. In this paper, focus is placed on acoustic wave transmission through a two-dimensional waveguide with wall corrugations having the geometry of square waves. The work intends to describe the acoustic field in terms of pressure and impedance along this type of periodic ducts, estimate the phase speed of transmitted wave at frequencies which lies within passbands, and compare the speed of sound in this type of waveguides to that in a duct with sinusoidal corrugations. Considering small corrugation amplitude, a small parameter is defined from the ratio of the corrugation amplitude and the waveguide average thickness and the perturbation method of strained parameters is used up to the second order. The influence of wall corrugation density is considered as a differentiating factor for graphical presentation of obtained analytical results.

#### 2. **Mathematical Model and Solution**

A two-dimensional acoustic waveguide (duct) is confined between two rigid walls having square-wave geometries, as shown in Figure 1. The mass velocity of air contained in the waveguide is much smaller than the speed of sound in this inviscid medium. Acoustic waves are considered in the form of perturbations of the medium pressure , density , and mass velocity , where is the time coordinate. The linearized continuity and momentum conservation equations are supplemented with a barotropic fluid equation of state to yield the following wave equations:where is the speed of sound, () represent the spatially uniform state, and is the specific heat ratio of the isentropic gas.