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.

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 [

After 1990, Bradley [

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.

A two-dimensional acoustic waveguide (duct) is confined between two rigid walls having square-wave geometries, as shown in Figure

The two-dimensional periodic acoustic duct.

Based on momentum conservation, a velocity potential

Assume that the acoustic waves along the duct propagate as time-harmonic signals such that

The inhomogeneous boundary conditions (

In order to solve the system of (

As with any perturbation technique, a systematic procedure for determining successively more accurate approximations is followed. Starting with the zeroth-order problem, let us select a mode with a frequency that lies in the passband part of the banded spectrum. Then, a solution of (

Next, let us solve the first-order problem. A particular solution for

The governing equation (

To solve the second-order problem, let us seek a solution for

A solution of the system of (

In this section, let us consider two acoustic waveguides with rigid walls having square-wave profiles. Both waveguides are assumed to be filled with stationary air, in which the bulk velocity of sound equals 343 m/s. Both periodic ducts have an average half width (

Figure

Pressure distribution of the fundamental mode at 1 kHz along two acoustic waveguides at

Specific impedance components (resistance and reactance) for the fundamental mode at 1 kHz along two acoustic waveguides, one with

Figure

Phase speed of the fundamental mode of two acoustic waveguides, one with

A further case to consider is a comparison between the acoustic phase speed in a duct having square-wave corrugation and that in a duct having sinusoidal corrugation with the same amplitude of the wall wavelength. Figure

Phase speed of the fundamental mode inside two acoustic waveguides, one with sinusoidal wall corrugations having

Acoustic waveguides with hard-wall corrugations having square-wave profile were considered. Since the amplitude of corrugation is small compared to the width of the duct, a small geometric parameter-based perturbation approach was used to describe the acoustic pressure inside the waveguide and to estimate phase speed of acoustic transmitting waves. Acoustic pressure and specific impedance along these waveguides were found to be periodic. The wall corrugation density was found to influence pressure inside periodic waveguides and phase speed of waves transmitting through them. A waveguide with low wall corrugation density is found to have significantly higher pressure values and lower phase speeds than those inside a waveguide with lower wall corrugation density. The phase speeds of waves transmitting in a duct with square-wave wall corrugations were found to be slower than phase speeds of waves transmitting through a duct with sinusoidal wall corrugations.

The author declares that there is no conflict of interests regarding the publication of this paper.

Research support offered by King Fahd University of Petroleum and Minerals (KFUPM) is acknowledged.