Multiple scattering between cylinders and a Schroeder diffuser

Chapter 12

See discussions, stats, and author profiles for this publication at:
https://www.researchgate.net/publication/233707635

Multiple Scattering Between Cylinders and a Schroeder Diffuser

Article in Acta Acustica united with Acustica $\cdot$ March 2010
DOI: 10.3813/AAA. 918278

CITATION
1

READS

64

6 authors, including:

Some of the authors of this publication are also working on these related projects:

Doped YIG and ZnO with rare earth as well as Co and also theoretical studies of liquid crystal and polymer ordering View project

Multiple Scattering Between Cylinders and a Schroeder Diffuser

M. A. Pogson ${ }^{11}$, D. M. Whittaker ${ }^{11}$, G. A. Gehring ${ }^{11}$, R. J. Hughes ${ }^{21}$, J. A. S. Angus ${ }^{21}$, T. J. Cox ${ }^{21}$
${ }^{1)}$ University of Sheffield, Sheffield, United Kingdom. markpogson@gmail.com
${ }^{2)}$ University of Salford

Abstract

Summary A multiple scattering method is developed to model an array of $N$ cylinders positioned in front of a Schroeder diffuser. Results are compared against the more computationally intensive boundary element method, as well as existing multiple scattering methods for cylinders alone. An investigation of scattering performance from a cylinder arrangement in front of a Schroeder diffuser is also performed. The addition of cylinders is shown to offer potential benefits such as reducing the effects of periodicity in the Schroeder diffuser.

PACS no. 43.20.Bi, 43.20.Fn, 43.55.Br

1. Introduction

Diffusers are employed in listening spaces to treat various acoustic problems, including echoes, while minimising removal of sound energy [1]. Diffusers should contain roughness at different scales to scatter sound across a wide bandwidth, aiming to provide predictable spatial and temporal dispersion of sound. Existing diffuser designs are principally surface-based, eminent among which is the phase grating Schroeder diffuser. This consists of a sequence of wells of depths determined by a pseudorandom number sequence [2].

In this paper we consider an array of cylinders introduced in front of a Schroeder diffuser in order to investigate potential improvements to the performance of the diffuser. Previous investigation of acoustic scattering from cylinder arrays has concentrated on sound attenuation or reduced transmission at selected frequencies [3, 4], whereas the intention here is for the cylinders to behave as a volumetric diffuser [5]. This offers potential benefits over surface diffusers in low frequency diffusion due to multiple scattering paths between elements. Certain arrangements of cylinders in front of a Schroeder diffuser may also reduce the effects of lobes and ‘plate’ frequencies (where the diffuser acts as a flat plate) associated with designs based on integer number sequences [1].

To facilitate investigation, a multiple scattering method for cylinders is extended to include multiple scattering from a Schroeder diffuser, although the method is applicable to any rough surface which may be approximated as a sequence of staggered slats. Multiple scattering methods require the scattered pressure from each scattering element

^[1]to be re-expanded in terms of coordinates centred on every other scattering element. While exact re-expansion of scattered waves between cylinders is possible [6], consideration of scattering between cylinders and the Schroeder diffuser is more problematic due to the different shapes of scattering elements and wave forms. Scattered waves are therefore approximated as planar on incidence at other scattering elements, which makes re-expansion of waves possible, similar to previous methods for scattering from cylinders alone $[7,8]$.

We investigate the accuracy of the present method by comparing results against BEM for a regular array of cylinders in front of a Schroeder diffuser. Although BEM is able to calculate scattering for arbitrary surface shapes, a multiple scattering approach offers advantages in terms of computational efficiency. A disordered array is subsequently considered to investigate scattering performance, which demonstrates significant benefits of the use of cylinders. The multiple scattering method is shown to be both efficient and accurate.

2. Formulation

2.1. General description

Multiple scattering of sound waves is considered between $N$ cylinders and a Schroeder diffuser comprising $N^{\prime}$ wells, thus solving the Helmholtz equation for complex acoustic pressure exterior to the scattering elements. The cylinders and wells are rigid, of infinite length and oriented in the same direction, as shown in Figure 1; the problem is therefore reduced to two-dimensions [9].

The plane wave approximation matches the complex pressure of a scattered wave at the centre of each element at which it is incident, with the angle of propagation given by the angle between elements. The plane wave

Figure 1. Cylinders and Schroeder wells. Polar coordinates centred on scattering element $u$ are denoted by $\left(r_{u}, \theta_{u}\right)$, and the position of $v$ in terms of polar coordinates centred on $u$ is denoted by ( $R_{u v}, \alpha_{u v}$ ); examples of these are shown. Cylinder $u$ is of radius $a_{u}$. Primed indices denote wells; well $u^{\prime}$ is of depth $d_{u^{\prime}}$ and all wells are the same width, $2 b$. Without loss of generality, the Schroeder diffuser is perpendicular to the $x$-axis.
approximation for cylinders has been shown to compare well with exact multiple scattering but with the advantage of being readily applicable to different scattering shapes and incident wave forms [10]. Although the approximation works best for large spacing between scattering elements, results remain accurate for smaller spacing between cylinders $[7,6]$.

The plane wave treatment of cylinders by McIver and Evans [7] is formulated as an $O\left(N^{4}\right)$ method; however, by following a similar approach to that for exact multiple scattering [6] we are able to reduce this to $O\left(N^{2}\right)$. Schroeder wells are added to the result without increasing the number of equations by considering scattering from the diffuser in terms of additional paths between cylinders. This neglects scattering directly between wells, which is acceptable given that scattering from the Schroeder diffuser is predicted with reasonable accuracy by assuming Kirchoff boundary conditions, where only the incident wave is scattered [1].

Each Schroeder well is treated separately to improve local plane wave approximation, with scattering calculated by a Fraunhofer method. This relies on a simplified integral of the Kirchoff boundary conditions, assuming far field scattering, which is consistent with both the approximation between cylinders and the original Schroeder design method [1].

The Kirchoff assumption requires small wavelength relative to surface width, which does not hold below approximately 500 Hz for typical 5 cm well width devices; since this is the lower limit of most diffuser frequency ranges, the approximation is fair. The method deteriorates for oblique angles and rapidly changing impedances, while the short wavelength assumption of the Kirchoff boundary conditions also conflicts to some extent with the long wavelength assumption of far field scattering [1]. These effects will be considered in the comparison of results.

By treating primary scattering from wells as an addition to the source wave, all higher orders of scattering
from wells can be modelled simply as additional scattering paths from cylinders. Hence the multiple scattering problem is expressed similarly to that for cylinders alone, with the total pressure $p$ at any point described by the sum of the source wave $p_{I}$ (including primary scattering from the Schroeder diffuser) plus the sum of scattered waves due to all orders of scattering $p_{X}^{\mathrm{s}}$ from each cylinder $u$ (including scattering paths via all wells),

$$p=p_{I}+\sum_{u=1}^{N} p_{X}^{\mathrm{s}}$$

All orders of multiple scattering between elements are thus obtained, neglecting only scattering directly between wells. To solve the multiple scattering problem, the source wave is first defined, along with the form of scattered waves. Expressions for scattered waves depend on the total incident pressure at each scattering element, which is evidently dependent on scattering from all other elements. Solution therefore requires simultaneous evaluation of the boundary conditions at each element, which is enabled by wave re-expansion.

2.2. Wave definitions

The pressure from the source reaching cylinder $v$ is

$$P_{0 v}=q \mathrm{e}^{\mathrm{i} k\left(x, \cos (\beta)+y, \sin (\beta)\right)}$$

where $k$ is the wavenumber, $\left(x_{v}, y_{v}\right)$ are the Cartesian coordinates of the centre of cylinder $v$, and the planar source wave propagates at angle $\beta$, with pressure $q$ at the origin.

Including scattering paths via all wells, the scattered pressure from cylinder $u$ is expressed in terms of a modal series [7] by

$$\begin{aligned} & p_{X}^{\mathrm{s}}\left(r_{u}, \theta_{u}\right)=\sum_{m=-\infty}^{\infty} A_{m}^{\mathrm{s}} L_{m}^{\mathrm{s}}\left[H_{m}\left(k r_{u}\right) \mathrm{e}^{\mathrm{i} m \theta_{u}}\right. \\ & \left.\quad+\sum_{u^{\prime}=1}^{N^{\prime}} H_{m}\left(k R_{u u^{\prime}}\right) \mathrm{e}^{\mathrm{i} m \alpha_{u u^{\prime}}} S_{u^{\prime}}\left(r_{u^{\prime}}, \theta_{u^{\prime}}, \alpha_{u u^{\prime}}\right)\right] \end{aligned}$$

where $A_{m}^{\mathrm{s}}$ are coefficients to be determined by simultaneous evaluation of boundary conditions, angles $\theta$ and $a$ and distances $r$ and $R$ are defined in Figure 1, primed indices denote wells, $S_{u^{\prime}}$ is a scattering function from well $u^{\prime}$ (defined below), $H_{m}$ is the Hankel function of the first kind of order $m$, and $L_{m}^{\mathrm{s}}$ is introduced for later convenience, defined for a rigid cylinder of radius $a_{u}$ as

$$L_{m}^{\mathrm{s}}=\frac{J_{m}^{\prime}\left(k a_{u}\right)}{H_{m}^{\prime}\left(k a_{u}\right)}$$

where $J_{m}$ is the Bessel function of the first kind of order $m$, and primes denote differentiation with respect to the argument.

The Schroeder diffuser is modelled as a diffraction grating with phase shifts introduced according to the depth of each well; no further consideration is made of the sides of wells and zero sound pressure is assumed behind wells.

The scattering function from well $u^{\prime}$, including an obliquity factor $[1,11]$, is therefore given by

$$\begin{aligned} S_{u^{\prime}}\left(r_{u^{\prime}}, \theta_{u^{\prime}}, \theta_{0}\right)= & b\left(1-\cos \theta_{u^{\prime}}\right) \mathrm{e}^{\mathrm{i} k\left(r_{u^{\prime}}+2 d_{u^{\prime}}\right)} \\ & \cdot \operatorname{sinc}\left(b k\left(\sin \theta_{u^{\prime}}-\sin \theta_{0}\right)\right) \end{aligned}$$

where $d_{u^{\prime}}$ is the depth of well $u^{\prime},\left(r_{u^{\prime}}, \theta_{u^{\prime}}\right)$ are polar coordinates centred on the face of well $u^{\prime}, \operatorname{sinc}(x)$ represents $\frac{\sin x}{2}$, and $\theta_{0}$ is the angle of propagation of the incident plane wave. Since the method is devised to model a Schroeder diffuser, all wells are assumed to be the same width $2 b$, although the individual width of well $u^{\prime}$ could be substituted in the scattering function were this not the case.

With incident pressure obtained as in equation (2), primary scattering from well $u^{\prime}$ reaching cylinder $v$ is given by

$$P_{0 v}^{u^{\prime}}=q \mathrm{e}^{\mathrm{i} k\left(x_{u^{\prime}} \cos (\beta)+y_{u^{\prime}} \sin (\beta)\right)} S_{u^{\prime}}\left(R_{u^{\prime} v}, \alpha_{u^{\prime} v}, \beta\right)$$

where $\left(x_{u^{\prime}}, y_{u^{\prime}}\right)$ are the Cartesian coordinates of the centre of the face of well $u^{\prime}$.

3. Multiple scattering solution

3.1. Re-expansion of waves

The sum of the source wave in equation (2) and primary scattering from wells in equation (6) can be re-expanded [7] in terms of polar coordinates centred on cylinder $v$ as

$$\begin{aligned} p_{I}\left(r_{v}, \theta_{v}\right)= & P_{0 v} \sum_{m=-\infty}^{\infty} J_{m}\left(k r_{v}\right) \mathrm{e}^{\mathrm{i} m\left(\frac{x}{2}-\theta_{v}-\beta\right)} \\ & +\sum_{u^{\prime}=1}^{N^{\prime}} P_{0 v}^{u^{\prime}} \sum_{m=-\infty}^{\infty} J_{m}\left(k r_{v}\right) \mathrm{e}^{\mathrm{i} m\left(\frac{x}{2}-\theta_{v}-\alpha_{u^{\prime}}\right)} \end{aligned}$$

where primary scattering from well $u^{\prime}$ is approximated as planar on incidence at cylinder $v$.

Applying the plane wave approximation to scattering between cylinders, the scattered wave from cylinder $u \neq v$ can be similarly re-expanded around cylinder $v$. The scattered wave is treated as $N^{\prime}+1$ separate plane waves incident at $v$ due to all possible scattering paths (via each well plus direct), which are re-expanded as

$$\begin{aligned} p_{S}^{u}\left(r_{v}, \theta_{v}\right)= & P_{u v} \sum_{m=-\infty}^{\infty} J_{m}\left(k r_{v}\right) \mathrm{e}^{\mathrm{i} m\left(\frac{x}{2}-\theta_{v}-\alpha_{u v}\right)} \\ & +\sum_{u^{\prime}=1}^{N^{\prime}} P_{u v}^{u^{\prime}} \sum_{m=-\infty}^{\infty} J_{m}\left(k r_{v}\right) \mathrm{e}^{\mathrm{i} m\left(\frac{x}{2}-\theta_{v}-\alpha_{u^{\prime}}\right)} \end{aligned}$$

where $P_{u v}$ is the directly scattered pressure from cylinder $u$ at cylinder $v$, and $P_{u v}^{u^{\prime}}$ is the scattered pressure from cylinder $u$ at cylinder $v$ via well $u^{\prime}$. From equation (3), separating terms for scattering via well $u^{\prime}$ and directly between cylinders, these values are

$$\begin{aligned} & P_{u v}=\sum_{m=-\infty}^{\infty} A_{m}^{u} L_{m}^{u} H_{m}\left(k R_{u v}\right) \mathrm{e}^{\mathrm{i} m \alpha_{u v}} \\ & P_{u v}^{u^{\prime}}=\sum_{m=-\infty}^{\infty} A_{m}^{u} L_{m}^{u} H_{m}\left(k R_{u u^{\prime}}\right) \mathrm{e}^{\mathrm{i} m \alpha_{u u^{\prime}}} S_{u^{\prime}}\left(R_{u^{\prime} v}, \alpha_{u^{\prime} v}, \alpha_{u u^{\prime}}\right) \end{aligned}$$

The first term of equation (8), describing direct scattering between cylinders, can alternatively be derived by exact re-expansion of the cylindrical scattered wave from $u$; the Bessel function contained in this result can then be asymptotically expanded for large $k r$, the first term of which corresponds to a plane wave [7]. The second term of the obtained series, a correction term to the plane wave approximation, is included by McIver and Evans [7], and can be straightforwardly included in the present method without affecting the overall procedure, but will be omitted for convenience and consistency with the approximation of scattering at wells.

Adding to equation (8) the scattered pressure from cylinder $v$ itself (obtained from equation (3), including scattering via the Schroeder diffuser), the overall scattered pressure is expressed around cylinder $v$ as

$$\begin{aligned} \sum_{u=1}^{N} p_{S}^{u}\left(r_{v}, \theta_{v}\right)= & \sum_{m=-\infty}^{\infty} A_{m}^{v} L_{m}^{v} H_{m}\left(k r_{v}\right) \mathrm{e}^{\mathrm{i} m \theta_{v}} \\ & +\sum_{u=1}^{N} P_{u v} \sum_{m=-\infty}^{N} A_{m}^{u} L_{m}^{u}\left(k r_{v}\right) \mathrm{e}^{\mathrm{i} m\left(\frac{x}{2}-\theta_{v}-\alpha_{u v}\right)} \\ & +\sum_{u=1}^{N} \sum_{u^{\prime}=1}^{N^{\prime}} P_{u v}^{u^{\prime}} \sum_{m=-\infty}^{\infty} J_{m}\left(k r_{v}\right) \mathrm{e}^{\mathrm{i} m\left(\frac{x}{2}-\theta_{v}-\alpha_{u^{\prime}}\right)} \end{aligned}$$

3.2. Evaluation of boundary conditions

The multiple scattering solution is obtained by simultaneous evaluation of the boundary conditions at each surface. Since scattering from the Schroeder diffuser is modelled in terms of additional scattering paths between cylinders, only boundary conditions at cylinders must be considered explicitly. For rigid cylinders, radial velocity on the surface is zero, hence at each cylinder $v$

$$\left.\sum_{u=1}^{N} \frac{\partial p_{S}^{u}}{\partial r_{v}}\right|_{r_{v}=\alpha_{v}}=\left.\frac{\partial p_{I}}{\partial r_{v}}\right|_{r_{v}=\alpha_{v}}, \quad v=1, \ldots, N$$

Differentiating equations (7) and (11) with respect to $r_{v}$ enables evaluation of equation (12); substituting in equations (9) and (10), rearranging, and truncating infinite sums to a finite limit $M$ to permit computation, gives the boundary condition at each cylinder $v$ as

$$\begin{aligned} A_{m}^{v}+ & \sum_{u=1, n \neq v}^{N} \sum_{n=-M}^{M} A_{n}^{u} L_{m}^{u} H_{n}\left(k R_{u v}\right) \mathrm{e}^{\mathrm{i}((n-m) \alpha_{u v}+m \frac{x}{2})} \\ + & \sum_{u=1}^{N} \sum_{u^{\prime}=1}^{N^{\prime}} \sum_{n=-M}^{M} A_{n}^{u} L_{m}^{u} H_{n}\left(k R_{u u^{\prime}}\right) \\ & \cdot S_{u^{\prime}}\left(R_{u^{\prime} v}, \alpha_{u^{\prime} v}, \alpha_{u u^{\prime}}\right) \mathrm{e}^{\mathrm{i}\left(n \alpha_{u u^{\prime}}+m\left(\frac{x}{2}-\alpha_{u^{\prime}}\right)\right)} \\ = & -P_{0 v} \mathrm{e}^{\mathrm{i} m\left(\frac{x}{2}-\beta\right)}-\sum_{u^{\prime}=1}^{N^{\prime}} P_{0 v}^{u^{\prime}} \mathrm{e}^{\mathrm{i} m\left(\frac{x}{2}-\alpha_{u^{\prime}}\right)}, \\ & v=1, \ldots, N \\ & m=-M, \ldots, M \end{aligned}$$

Solution of the simultaneous equations of equation (13) for all $v$ and $m$ yields every $A_{m}^{v}$ value, equivalent to previous multiple scattering solutions [6]. Substituting results for $A_{m}^{v}$ into equation (3) provides the scattered pressure at any point, which can be added to the primary scatter from the Schroeder diffuser to give the total scattered pressure.

4. Results and discussion

4.1. Notes on method

By setting all $S_{n^{\prime}}$ values to zero, multiple scattering in the absence of the Schroeder diffuser is obtained. With the Schroeder diffuser removed in this way, results from the present method are identical to those obtained by McIver and Evans [7] in the absence of the plane wave correction term. As stated for equation (8), this term can be added to the current method without increasing the number of equations, again obtaining identical results to McIver and Evans.

The method of McIver and Evans was formulated as a set of $N(N-1)$ equations. Following a similar procedure to that for exact multiple scattering [6], the present method simplifies the approximation to $N(2 M+1)$ equations, where $M$ is the highest scattering mode considered for each cylinder. This is achieved by solving for the scattered pressure at each cylinder from all other cylinders without explicit distinction of the contribution of each cylinder. The amount of computation is therefore significantly reduced for $N>M$, which is typically the case. Schroeder wells are added without increasing the number of equations.

The value of $M$ is chosen such that further terms of the infinite series do not affect the result at the required precision; the value is often small and is independent of the number of cylinders. In the case of $k a \rightarrow 0$, which holds for many important frequencies for room acoustics with cylinder diameters $<10 \mathrm{~cm}$ [12], the method is particularly efficient since terms for $n, m>1$ are vanishingly small [13], hence only $3 N$ equations are required.

It is possible to include the effects of an infinite wall on which the Schroeder diffuser may be positioned. This is achieved by the method of images [14] in combination with altering the well scattering function to be the difference between scattering from well $u^{\prime}$ and a section of flat wall in the same position, hence effectively removing the wall from where the Schroeder diffuser is positioned. Since nothing more is contributed to the method, further discussion is omitted and effects not included in results.

4.2. Comparison with other methods

To assess the accuracy of the present method, results for a straightforward array of scattering elements are compared against appropriate existing methods. A more complicated array is subsequently considered to investigate the benefits of cylinders to diffuser performance, as well as limits of the model.

Figure 2. Scattering arrangement used for comparison of methods. Cylinder radii 5 cm , well widths 10 cm . Source wave incident from left to right.

Results from BEM are treated as correct for the purposes of comparison, as the technique is highly accurate and has been validated by practical experiment [1, 9]. Computation for BEM is $O\left(N^{2}\right)$, as for the present method. However, the greater number of procedures involved in BEM, such as performing surface integrals, requires more computation. The value of $N$ is also comparatively larger in BEM due to the number of mesh points required per scattering element, which increases with frequency and is typically at least 8 per cylinder. More significantly, because the present method does not require additional equations for Schroeder wells, its comparative efficiency is significantly increased.

Polar plots of the magnitude of scattered pressure are shown for both the present method and BEM. The diffusion coefficient, an averaged circular autocorrelation of the scattered pressure [1], is also used to compare results; the metric allows clear comparison between methods, and is an important measure of the effectiveness of diffusers [15], so must be well approximated for the method to be successful. The diffusion coefficient ranges from 0 to 1 , with better diffusion being indicated by larger values, and is defined as
$d=\left(\left(\sum_{j=1}^{n}\left|p_{j}\right|^{2}\right)^{2} / \sum_{j=1}^{n}\left|p_{j}\right|^{4}-1\right) /(n-1)$.
where $n$ is the number of evenly spaced receiver positions, and $p_{j}$ is the scattered pressure at receiver $j$.

A regular 3-by-3 array of cylinders is positioned in front of a Schroeder diffuser of design frequency 500 Hz , with well depths defined by a quadratic residue sequence of length 7 , and well widths of 10 cm , as shown in Figure 2. The cylinders are all of radius 5 cm with centre-tocentre spacing of 20 cm , and the central cylinder is 50 cm from the centre of the Schroeder diffuser face. Receivers form a semicircle of radius 5 m centred on the face of the Schroeder diffuser, and the source wave propagates normal to the Schroeder diffuser.

Figure 3. Comparison of methods for cylinders. Polar plots of the magnitude of scattered pressure show results from the plane wave multiple scattering of the present method (a) and BEM (b); corresponding diffusion coefficients are also shown ©, including results for exact multiple scattering.

In addition to results for the whole array, scattering from the cylinders and the Schroeder diffuser are calculated separately in order to investigate the method in greater detail.

Figure 3 shows results for scattering from the cylinders alone in free space, with the array centred on the origin for consistency with results from the Schroeder diffuser. Polar plots for the magnitude of scattered pressure are shown for the plane wave multiple scattering of the present method (a) and BEM (b). A graph of diffusion coefficients is also shown ©, which includes additional results for exact multiple scattering. For both multiple scattering methods, $M=3$. Good agreement is evident between all methods.

Results for scattering from the Schroeder diffuser are shown in Figure 4. Polar plots of scattering are shown for the Fraunhofer approximation used in the present method (a) and BEM (b), along with a graph of diffusion coefficients ©. Agreement is slightly worse than for cylinders due to greater approximation versus BEM, which explicitly models the sides of wells for example, but compari-

Figure 4. Comparison of methods for the Schroeder diffuser. Polar plots of the magnitude of scattered pressure show results from the Fraunhofer approximation used in the present method (a) and BEM (b); corresponding diffusion coefficients are also shown ©.
son remains reasonable. It is evident in the polar plots that low frequency scattering amplitude is over-estimated by the Fraunhofer method, where the Kirchoff assumption deteriorates, as discussed above. However, the polar pattern remains similar, which is reflected in the similarity of diffusion coefficients.

Figure 5 shows results for the whole array. Polar plots from the present method (a) and BEM (b) compare reasonably, although over-estimation of low frequency scattering amplitude from the Schroeder diffuser is again evident. The graph of diffusion coefficients © includes results in the absence of multiple scattering between the Schroeder diffuser and cylinders (i.e. only multiple scattering between cylinders plus primary scattering from the Schroeder diffuser). This demonstrates that multiple scattering between the Schroeder diffuser and cylinders has a notable positive effect on agreement with BEM.

Because the cylinders are quite directly in front of the Schroeder wells, scattering between them does not suffer so much from the greater inaccuracy of the Fraunhofer

Figure 5. Comparison of methods for the whole array shown in Figure 2. Polar plots of the magnitude of scattered pressure show results from the present method (a) and BEM (b); corresponding diffusion coefficients are also shown ©, including results without multiple scattering from the Schroeder diffuser.
method at oblique angles. It is worth noting that the cylinders are in the near-field to the Schroeder diffuser across the frequency range, and results remain reasonably accurate at higher frequencies despite the extension of the near field. Although some discrepancies are evident in the polar plots, agreement between diffusion coefficients for the present method and BEM is good, which suggests the method is able to assess accurately the effectiveness of diffusers.

4.3. Further investigation of scattering

While results in Figure 5 demonstrate the accuracy of the present method, comparison with Figure 4 suggests that the presence of cylinders contributes little to the performance of the diffuser with the simple array considered. A more complicated arrangement is therefore used to investigate diffuser performance while further testing the present method.

A disordered array of cylinders is positioned in front of a periodically repeated Schroeder diffuser, as shown

Figure 6. Scattering arrangement used for further investigation. Cylinder radii either 2.5 cm or 1.25 cm , well widths 5 cm .
in Figure 6. The Schroeder diffuser is composed of five repetitions of the one shown in Figure 2, but with 5 cm well widths. The cylinders are either 2.5 cm or 1.25 cm in radius, symmetrically occupying 16 positions on a 2-by18 grid with 9 cm spacing between grid positions, which is incommensurate with the period of the Schroeder diffuser. Separation between the front of the Schroeder diffuser and the closer line of grid points is 25 cm . Receivers form a semicircle of radius 10 m centred on the face of the Schroeder diffuser, and the source wave propagates normal to the Schroeder diffuser.

The diffusion coefficient for the whole array is shown in Figure 7 (solid line), along with results for the Schroeder diffuser alone (dashed line), a flat plate of the same total width as the Schroeder diffuser (dash-dotted line), and cylinders positioned in front of the flat plate (dotted line). Results are shown for the present method (a) and BEM (b). Both methods give broadly similar results, with the presence of cylinders in front of the Schroeder diffuser providing a notable increase in the diffusion coefficient across the frequency range.

For simplicity, results for cylinders in free space are omitted as their diffusion is not directly comparable with other arrangements; results for cylinders in front of a plate are presented instead, providing fairer comparison of diffuser performance. Nonetheless, agreement between methods for cylinders in free space is very similar to before, and the diffusion coefficient is predominantly below that for the whole array.

Agreement with BEM for the Schroeder diffuser is slightly worse than previously, mainly due to the halving of well widths, making the limitations of the Kirchoff assumption in the present method more pronounced due to more rapid fluctuations of impedance across the diffuser, although the total width is greater. Results for the whole array compare reasonably with BEM; deterioration in agreement from previously is largely due to the noted worsening of the Schroeder result, in combination with smaller separation between scattering elements, and more oblique angles between Schroeder wells and cylinders. The same limitations apply to the result for cylinders in front of the

Figure 7. Diffusion coefficients from scattering arrangement shown in Figure 6. Multiple scattering (a), and BEM (b).
flat plate, which uses the same method but with well depths of zero.

Despite some discrepancies in the present method, prediction of improvements to diffuser performance by the inclusion of cylinders is clear. The plate frequency evident towards the top of the frequency range considered is effectively removed, and low frequency diffusion is improved. The latter is underestimated by the present method since the Fraunhofer approximation is prone to overestimating the magnitude of scattering from the Schroeder diffuser at low frequency, as discussed for Figure 4, hence dominating the result; this results in the diffusion coefficient being closer to that for the Schroeder diffuser alone. However, the scattering arrangement was deliberately chosen to investigate limits of the present method, and results remain accurate given the assumptions of the model, providing a good overall prediction of scattering.

Our results suggest that cylinders in free space can provide good diffusion in their own right, which may be desirable in certain settings, although consideration of effects from surroundings is important for a full evaluation. The addition of cylinders to a Schroeder diffuser has the advantage of providing greater back-scattered power than cylinders alone, as well as offering useful improvements to existing designs. Due to the viable increase in low frequency diffusion with cylinders, shallower Schroeder diffusers may be employed, offering clear practical benefits.

5. Conclusions and further work

A multiple scattering method has been developed to model a finite array of cylinders in front of a Schroeder diffuser. Scattering between cylinders replicates an existing plane wave approximation [7], but with the number of simultaneous equations significantly reduced. Multiple scattering effects of the Schroeder diffuser are added without increasing the number of equations; this is achieved by incorporating scattering from Schroeder wells into terms for scattering from cylinders, which is made possible by neglecting scattering directly between wells. Inclusion of the Schroeder diffuser in the multiple scattering method exploits the applicability of the plane wave approximation to different scattering shapes and wave forms. The efficiency of the method makes it suitable for use in optimisation procedures for diffuser design [1].

The present method compares favourably with BEM, performing well even in the near field, despite the far field assumptions of the model. Limits of the model are explored, and notable benefits of cylinder arrays in diffuser design are demonstrated.

As well as providing improvements to spatial diffusion, the use of cylinders should also increase temporal diffusion due the provision of multiple scattering paths between elements. Of interest for further investigation would be the use of non-grid spacing of cylinders to avoid periodicity, the effects of reflective walls, and practical positioning of cylinders for realistic application.

References

[1] T. J. Cox, P. D’Antonio: Acoustic absorbers and diffusers. 2nd ed. Spon Press, 2009.
[2] M. R. Schroeder: Diffuse sound reflection by maximumlength sequences. J. Acoust. Soc. Am. 57 (1975) 149-150.
[3] O. Umnova, K. Attenborough, C. M. Linton: Effect of porous covering on sound attenuation by periodic arrays of cylinders. J. Acoust. Soc. Am. 119 (2006) 278-284.
[4] J. V. Sanchez-Perez, D. Caballero, R. Martinez-Sala, C. Rubio, J. Sanchez-Dehesa, F. Meseguer, J. Llinarez, F. Galvez: Sound attenuation by a two-dimensional array of rigid cylinders. Phys. Rev. Lett. 80 (1998) 5325-5328.
[5] T. J. Cox, R. J. Hughes, J. A. S. Angus, D. M. Whittaker, M. Pogson, G. A. Gehring: Volumetric diffusers inspired by percolation fractals. Proceedings of the Institute of Acoustics 30 (2008).
[6] C. M. Linton, D. V. Evans: The interaction of waves with arrays of vertical cylinders. J. Fluid Mech. 215 (1990) 549569 .
[7] P. McIver, D. V. Evans: Approximation of wave forces on cylinder arrays. Appl. Ocean Res. 6 (1984) 101-107.
[8] M. J. Simon: Multiple scattering in arrays of axisymmetric waveenergy devices. part 1. a matrix method using a planewave approximation. J. Fluid Mech. 120 (1982) 1.
[9] T. J. Cox: Predicting the scattering from reflectors and diffusers using two-dimensional boundary element methods. J. Acoust. Soc. Am. 96 (1994) 874-878.

[10] S. A. Mavrakos, P. McIver: Comparison of methods for computing hydrodynamic characteristics of arrays of wave power devices. Appl. Ocean Res. 19 (1997) 283-291.
[11] E. Hecht: Optics. 4th ed. Addison-Wesley, 2001.
[12] H. Kattruff: Room acoustics. 4th ed. Spon Press, 2000.
[13] P. M. Morse, K. U. Ingard: Theoretical acoustics. McGrawHill, 1968.
[14] R. P. Feynmann, R. B. Leighton, M. Sands: The feynman lectures on physics. Vol. 2. Addison-Wesley, 1964.
[15] T. J. Cox, B.-I. L. Dalenback, P. D’Antonio, J. J. Embrechts, J. Y. Jeon, E. Mommertz, M. Vorländer: A tutorial on scattering and diffusion coefficients for room acoustic surfaces. Acta Acustica united with Acustica 92 (2006) $1-15$.