Proceedings of the Institute of Acoustics

 

 

Low frequency attenuation of acoustic waves using sound-soft scatterers

 

Alexander Dell¹, University of Sheffield, Sheffield, United Kingdom
Anton Krynkin, University of Sheffield, Sheffield, United Kingdom
Kirill Horoshenkov, University of Sheffield, Sheffield, United Kingdom
Gavin Sailor, University of Sheffield, Sheffield, United Kingdom

 

ABSTRACT

 

The attenuation of acoustic waves by silencers is typically achieved through the employment of rigidly backed cavities, connected to a main waveguide by a perforated panel. This invokes a resonant response at frequencies determined by the dimensions of the perforation and the rigidly backed cavity. A limiting factor in this approach is that to achieve low frequency attenuation, either large cavity depths are required, which is often impractical, or narrow neck regions need to be used, where performance is limited due to the viscothermal losses. Within this paper, a non-rigidly backed perforated pipe is presented where the pressure release condition at each perforation creates an acoustic sound-soft boundary condition. By the introduction of sound-soft boundary conditions, a low frequency band gap is created as the first order mode is shifted to a non-zero value, the frequency of which is determined by the dimensions and separation of the perforations. Experimental results displaying this band gap are presented along with a numerically validated ideal analytical model based upon the transfer matrix method.

 

1. INTRODUCTION

 

The attenuation of acoustic waves by silencers is achieved through the employment of rigidly backed cavities, connected to a main waveguide by a perforated panel. For silencers with partitioned cavities with a single perforation, i.e. a Helmholtz resonator, excellent attenuation can be achieved at the resonant frequency of the resonator. For silencers with un-partitioned cavities and panels composed of multiple identical perforations along the length of the silencer, a similar phenomenon occurs. If the perforated separating panel has a low porosity, a Helmholtz resonator type of attenuation occurs. As the porosity increases, the silencer behaves more like an expansion chamber [1]. Different configurations of partitions can be used to alter the type and number of resonances and dissipative materials can be introduced to help achieve more broadband attenuation [2]. A limitation of these types of silencers is the requirement for large cavity volumes, or narrow neck regions, to achieve low frequency attenuation. A large cavity volume is often impractical for engineering applications and having narrow regions often results in poor attenuation of acoustic waves due to the incurred viscothermal losses.

 

It has been theoretically shown that a sonic crystal in which the surfaces of each scatterer are modelled as sound-soft boundary conditions exhibit a non-zero first order mode [3]. Within this paper, an ideal analytical model based upon the transfer matrix method is presented. This model is validated for various geometrical configurations. Experimental evidence of a band gap produced by sound soft scatterers is also presented. This is for a non-rigidly backed perforated pipe in which the pressure release condition at each perforation creates an acoustic sound-soft boundary condition, thus turning each perforation into a sound-soft scatterer.

 

1.1. Finite periodic system of sound soft scatterers

 

The transfer matrix method (TMM) provides the relationship between the initial sound pressure, p, and volume flux, V = vS_a , where S_a is the cross sectional area and v is the acoustic particle velocity, at the start (x = 0) and at the end (x = L) of a medium in a duct [4]. To differentiate between the initial and end properties, the subscripts 0 and L are used, respectively. The transfer matrix, T, is derived under the assumption that only plane waves propagate through the medium in the x direction, meaning it provides the solution for a 1D wave propagation problem [5]. The general formulation of the transfer matrix is as follows;

 

 

The transfer matrix for a single fluid layer is constructed as

 

 

Here, Z = ρc/S_a is the characteristic impedance, k is the acoustic wavenumber and L is the length of the fluid layer.

 

Consider a finite waveguide with n periodic arrangements of sound soft scatterers. Each unit cell has a length h. The general geometry of the system can be seen in Figure 1.

 

The transfer matrix for the total system is given by

 

 

 

Figure 1: Graphical representation of a finite system of sound soft scatterers.

 

M_w  is the transfer matrix for a fluid layer of length h/2 and M_s  is given by

 

 

where N denotes the number of scatterers per unit cell. The total length of the system L = nh. The characteristic impedance of a perforation with an acoustic soft boundary, Z_s , is

 

 

where Z_p is the characteristic impedance, and k_p is the acoustic wavenumber, of the fluid within the perforation. Additionally, ∆l  is the length correction due to the pressure discontinuity between the perforation and waveguide, which is given by [6]

 

 

Here r_w is the hydraulic radius of the waveguide and r_p is the radius of the perforation. The transmission loss (TL) of the total system can then be obtained as

 

 

where Z is the characteristic impedance of the fluid within the waveguide. As this system is symmetric and isotropic, the transmission, reflection and absorption coefficients can be determined as

 

 

1.2. Bloch waves in an infinitely periodic structure

 

Assuming plane wave propagation in a periodic waveguide, the Bloch Floquet theorem can be fulfilled such that the forward and backward propagating Bloch waves display the same dispersion. The Bloch dispersion relation can be found as [7]

 

 

where q is the Bloch wavenumber and T is the transfer matrix for a single unit cell. The dispersion relationship for a periodic array of sound-soft backed perforations can then be expressed as

 

 

2. ANALYTICAL AND NUMERICAL RESULTS

 

Here the ideal model analytical model is validated using 3D numerical models computed in COMSOL Multiphysics 5.9. Four different sets of geometrical parameters were used to determine the relationship between the separation, depth and area of the perforations on the size and amplitude of the band gap produced. Details of the selected geometries can be seen in Table 1. In all cases the radius of the waveguide, r_w, was 40 mm, the total number of unit cells, n, was 10 and the number of perforations per unit cell, N, was 6. These perforations were distributed evenly along the circumference of the waveguide.

 

Table 1: Geometrical parameters of the four numerically validated models.

 

 

From Figure 2a, it is evident that the introduction of sound-soft backed perforations facilitates a reflection coefficient of near unity at 0 Hz, which then extends to approximately 300 Hz before dropping off. Consequently, the transmission coefficient behaves in an opposite manner, with a near zero value at 0 Hz before increasing to near unity when the reflection coefficient reaches zero. A band gap is produced within this frequency range. In Figure 2b this band gap is evident, denoted by the purely imaginary Bloch wavenumber. Physically, this purely imaginary Bloch wavenumber indicates that the propagating wave within the the band gap frequency range is evanescent. With each subsequent unit cell, the propagating waves is attenuated such that in an infinitely periodic system, no acoustic waves can propagate. In a finite system, this effect is still present, but the acoustic waves are not fully attenuated, indicated by the transmission loss plots in 2b.

 

In Model 2, the length of the unit cell has been halved, reducing the separation between the perforations. When examining 2c, the same phenomenon as in Model 1 is evident. The main difference being that by decreasing the unit cell length, the size of the band gap is almost doubled. It is worth noting that the reflection and transmission coefficients appear to change a lot more gradually in this scenario. When looking at Figures 2b and 2d, the transmission loss is almost identical in amplitude. This indicates that whilst the separation distance controls the size of the band gap, it is the number unit cells and perforations per unit cell which control the amplitude of attenuation.

 

To further establish the relationship between the perforation geometry and the size of the band gap produced, Model 3 has the same geometrical parameters as Model 1, apart from an increase of perforation radius by 1 mm. When examining Figure 2e, the main difference to Model 1 is, again, an increase in the size of the band gap. This time, the change in reflection and transmission coefficients is similar to that produced by Model 1. From Figure 2f it can be seen that by increasing the radius of the perforations, not only do you increase the size of the band gap, but you also increase the amount of attenuation within the band gap. This indicates that the overall surface area with soft boundary conditions has a direct affect on the amount of attenuation achieved.

 

The final parameter to examine in isolation is how the depth of the perforation influences the size and amplitude of the band gap produced. In Model 4, the depth is doubled, whilst the remaining parameters are kept consistent with Model 1. From Figure 2g, it can be seen that the size of the band gap is reduced, with the reflection and transmission coefficients changing more gradually, similar to that of Model 2. From Figure 2h, it can be seen that by increasing the depth of the perforations, the amount of attenuation is reduced, as evidenced by the reduction in transmission loss in comparison to Model 1. It can therefore be determined that to increase the amount of attenuation within a band gap, it is preferable to have perforations with less depth.

 

It can therefore be determined that the size and strength of the band gap is dependent on multiple factors. By reducing the unit cell length, but keeping the total number of unit cells, you will increase the width of the gap, but not increase the attenuation within the band gap. By increasing the size of the perforation, you are increasing the surface area where a sound-soft boundary condition is present, which will increase the size and attenuation within the band gap. Finally, you can adjust the size and attenuation of the band gap by changing the depth of the perforations, but an increase of depth will results in a reduction in the size and attenuation achieved by the band gap. In all cases excellent agreement with the numerical results are found, validating the ideal analytical model.

 

Figure 2: Analytical and numerical plots of the absorption, reflection and transmission coefficients for models 1 - 4; paired with plots of the real and imaginary components of the Bloch wavenumber against analytical and numerical plots of the transmission loss.

 

3. EXPERIMENTAL RESULTS

 

In order to validate the proposed analytical and numerical models and confirm the observed phenomenon, an experimental setup to investigate the sound propagation in a non-rigidly backed perforated pipe was designed. It involved a perforated pipe with three perforations per cross-section and 1 m of separation between the speaker and an array of 9 microphones, as shown in Figure 3 (a). The sound speaker was driven by a sine sweep generated between 50 Hz and 25 kHz. The speaker was installed on one side of the open ended perforated pipe as shown in Figure 3 (a). The microphone array was made of 9 GRAS 46AE 1/2” CCP Free-field Standard Microphone Sets and was arranged along the pipe diameter as shown in Figure 3 (b). It is noted that the location of holes in the perforated pipe along the pipe circumference were not consistent and varied along the pipe length. The spacing between the rows of perforation was fixed at 20 mm and the width of each perforation was approximately 25 mm as shown in Figures 4 (a) and (b). Figure 5 illustrates the transmission loss obtained with the following equation

 

 

where p_ref  is the reference microphone in the vicinity of the sound speaker and p_rec is the acoustic pressure recorded at the receiver side with the array microphones.

 

Figure 3: Experimental setup build at the Integrated Civil and Infrastructure Research Centre (ICAIR): (a) Sound speaker at the source end; (b) Array of 9 microphones at the receiver end; (c) Perforated pipe with speaker and array installed inside the pipe

 

 

Figure 4: Experimental pipe perforation: (a) Axial distance between perforation; (b) Width of a single perforation

 

Figure 5: Numerical and Experimental plots of the Transmission loss. The dashed line (- -) are the numerical results and the solid line (–) are the experimental results.

 

With the sound speaker designed to generate sound starting from 100 Hz, the experimental transmission loss offers the evidence that the periodically arranged holes in the rigid pipe create a low frequency band gap that starts from 0 Hz and ends at 460 Hz. Its width matches the predictions obtained for the simplified geometry where the perforations were idealised with an elliptical geometry in the 3D finite element model.

 

4. CONCLUSIONS

 

An ideal analytical model based upon the transfer matrix method has been developed to model the acoustic attenuation achieved by periodic arrays of non-rigidly backed perforations acting as sound-soft scatterers. It has been shown that periodic arrays of sound soft scatterers produce a low frequency band gap from 0 Hz to a frequency determined by the geometry of the perforations and the unit cell length. Acoustic waves within the frequency range of the bandgap become evanescent, achieving large amounts of attenuation for finite systems, and no wave propagation for infinite systems. By reducing the unit cell length, but keeping the total number of unit cells constant, you will increase the width of the gap, but not increase the attenuation within the band gap. By increasing the size of the perforation, you are increasing the surface area where a sound-soft boundary condition is present, which will increase the size and attenuation within the band gap. Additionally, you can adjust the size and attenuation of the band gap by changing the depth of the perforations, but an increase of depth will results in a reduction in the size and attenuation achieved by the band gap. All of these observations are numerically validated indicating in the ideal case the analytical model is valid.

 

In order to validate the sound-soft scatterer phenomenon observed in the analytical and numerical models, an experimental setup was design to investigate the acoustic wave propagation in a non-rigidly backed perforated pipe. The pipe had 3 perforations per cross section and there was a separation of 1 m between the speaker and array of 9 microphones. A 3D numerical model replicating the geometry of the perforated pipe was created to validate the experimental results. It can be seen form the results that the low frequency band gap is present, indicating that the non-rigidly backed perforations do indeed behave as sound-soft scatterers. There is good agreement between experimental and numerical results in regards to the frequency range in which a band gap is present and also the amplitude of the transmission loss. It is evident, however, that the experimental dataset is noisy. This could be attributed to a multitude of factors such as structure-borne vibrations or sound leakage due to the experimental perforated pipe being open-ended, with the experiment not taking place in an anechoic chamber. Despite this, it can be seen that the sound-soft scatterer phenomenon does occur when you have a pressure release condition from a non-rigidly backed perforation, allowing for very low frequency attenuation of acoustic waves with a sample size much smaller than the wavelength of sound being attenuated.

 

5. REFERENCES

 

1. Joseph W. Sullivan and Malcolm J. Crocker. Analysis of concentric-tube resonators having unpartitioned cavities. Journal of the Acoustical Society of America, 64(1):207, feb 1978.
2. I. Lee. Acoustic Characteristics of Perforated Dissipative and Hybrid Silencers. PhD thesis, The Ohio State University, 2005.
3. A. Krynkin and P. McIver. Approximations to wave propagation through a lattice of dirichlet scatterers. Waves in Random and Complex Media, 19(2):347–365, Jun 2009.
4. Noureddine Atalla Jean F. Allard. Propagation of Sound in Porous Media. John Wiley & Sons, 2009.
5. A. Dell, A. Krynkin, and K.V. Horoshenkov. The use of the transfer matrix method to predict the effective fluid properties of acoustical systems. Applied Acoustics, 182:108259, Nov 2021.
6. V Dubos, Jean Kergomard, Ali Khettabi, Jean-Pierre Dalmont, D.H. Keefe, and C.J. Nederveen. Theory of sound propagation in a duct with a branched tube using modal decomposition. Acta Acustica united with Acustica, 85:153–169, 03 1999.
7. C. E. Bradley. Time harmonic acoustic bloch wave propagation in periodic waveguides. part i. theory. The Journal of the Acoustical Society of America, 96(3):1844–1853, sep 1994.

 

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¹alexanderjdell1@gmail.com