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Numerical prediction of sound scattering from surfaces with fractal geometry: A preliminary investigation
Abstract
Sound diffusive and scattering surfaces can be implemented in architectural spaces to improve the acoustical qualities of the space, particularly by attenuating the effects of harsh reflections and by producing a more diffuse sound field. These surfaces typically are effective only for a limited range of frequencies, dependent on the scale of the surface geometry. Given the broad frequency range of human hearing, an ideal diffuser would provide scattering across many frequencies. There is a direct relationship between surface roughness size and the wavelength of the scattered sound; therefore, scale-invariant fractal surfaces can be useful in achieving this ideal. In this study, virtual 1-D fractal surfaces have been generated using the Random Midpoint Displacement (RMD) algorithm. A BEM method to simulate the sound diffusive properties of these surfaces was developed and some preliminary results are presented.
... While in the middle and high-frequency bands, the small diffuser trap depth gradually fulfills a role in the diffusion acoustic field. Bradley et al. (Bradley, Snow, Riegel, Nasipak, & Terenzi, 2011) generated a one-dimensional fractal surface through the random midpoint displacement (RMD) algorithm. It was tested using the boundary element method (BEM) simulation, confirming that the fractal interface had a higher scattering coefficient than the non-fractal interface. ...
Diffusive surfaces should be optimally designed for acoustic and aesthetic purposes. Fractals are commonly used to adapt to the parametric demands of interface design combining mathematical calculation and artistic creation. The author’s previous study proposed an improved fractal acoustic structure based on the Sierpinski-triangle building rule. However, there is insufficient research on quantifying the physiological and visual benefits of such fractal features. Focusing on fractal iteration and randomness of module arrangement as fractal parameters, this study investigates in 3D-virtual reality meeting rooms and identifies the influence of fractal diffusion structures on visual preferences and physiological responses. Variance analysis is used to construct models of fractal parameters, subjective preference factors, and physiological indicators. Results suggest that the Sierpinski-triangle pattern iterated twice with mid-high complexity (quantified by the fractal dimension D = 1.58) is significantly more visually preferred than other fractals. When the modules are at low randomness (quantified by P = 0.25), more random module arrangements elicited better beneficial outcomes. However, as the degree of randomness increased, effects became increasingly irregular. Subjective evaluation factors and physiological indicators show a strong correlation. To sum up, the use of the medium to mid-high complexity fractals in interior spaces would help enhance the visual preferences and stress relief of occupants.
... Lee et al. [25] showed that organizing modules in a fractal manner can improve the diffusion capacity of diffusers across a broad frequency range through a scale model experiment. Bradley et al. [26,27] proposed the virtual 1-D fractal surfaces generated using the Random Midpoint Displacement (RMD), the acoustic effect of fractal design parameters has been presented using the numerical predictions and experimental measurement. Perry et al. [28] developed a framework to optimize acoustic diffusers in a reasonable time without the need for boundary element predictions, and in the process, the modular fractal forms have been generated to extend the bandwidth. ...
This paper proposes a new design of an acoustic diffuser based on the construction rules of the Sierpinski triangle in order to broaden the effective diffusion frequency range. The diffuser is made of triangular blocks of different sizes attached to a plane surface. The effects of the number of fractal iterations, the height of triangular blocks, and arrangements of the blocks on the normal-incidence diffusion coefficients in the near field are examined through numerical simulations based on the boundary element method (BEM) in the frequency range of 100 Hz – 5 kHz. Furthermore, measurement results will be presented to validate the diffusion performance presented by the numerical simulations. The diffusion performance of a conventional quadratic residue diffuser (QRD) is compared to confirm the advantage of the designed diffuser for broadening the effective frequency range. It shows that the fractal patterns with various sizes of blocks improve diffusion performance compared to the conventional QRD of the same size, especially in the mid-low frequency range below 1 kHz.
In some situations, it is difficult to image seismic data even when factors such as illumination, anisotropy, and attenuation are resolvable. An often-overlooked but dominant factor affecting imaging is the geometric complexity of subsurface reflectors. In some of these settings, we propose that geometric complexity plays a dominant role in the quality of seismic imaging. This has been documented in subsalt regions with poor image quality where 3D vertical seismic profiles (VSPs) are available. VSPs often demonstrate that there is good illumination below salt, but the complexity of the observed downgoing wave and diffractions cannot be simulated with smooth earth models. We use synthetic examples with high geometric complexity and no other complications to analyze the imaging process: a 2D sediment-salt model with rough salt-top topography, and a 1D model with complex reflectivity. In these models, it is difficult to estimate velocity, and it is challenging to image the data. These synthetic examples are useful to understand the limitations of imaging with smooth models, to create guidelines to identify geometric complexity in field data, and to develop possible mitigations to improve imaging. Further real data examples will be needed to test geometric complexity.
A visual language for designs of acoustical arrays is defined by a shape grammar that deploys design rules using shape representations of acoustic panels with labels and weights to specify known formal and performative information. Reciprocal 3-D acoustic plots are computationally simulated using a perfectly matched layer finite element method and inform how the interactions of invisible sound waves are impacted by visual design decisions. This paper presents the results of this framework with 3 × 3 arrays of diffusers and absorbers, resulting in the introduction of an aesthetic value , incorporated with the diffusion coefficient, allowing for nuanced objective performative evaluation of these arrays. This ultimately leads to creative visual expressions of design rooted in acoustic theory and performance.
To study multi-fractal behavior of corroded steel surface, a range of fractal surfaces of corroded surfaces of Q235 steel were constructed by using the Weierstrass-Mandelbrot method under a high total accuracy. The multi-fractal spectrum of fractal surface of corroded steel was calculated to study the multi-fractal characteristics of the W-M corroded surface. Based on the shape feature of the multi-fractal spectrum of corroded steel surface, the least squares method was applied to the quadratic fitting of the multi-fractal spectrum of corroded surface. The fitting function was quantitatively analyzed to simplify the calculation of multi-fractal characteristics of corroded surface. The results showed that the multi-fractal spectrum of corroded surface was fitted well with the method using quadratic curve fitting, and the evolution rules and trends were forecasted accurately. The findings can be applied to research on the mechanisms of corroded surface formation of steel and provide a new approach for the establishment of corrosion damage constitutive models of steel.
A generalized Von Koch surface was constructed. On the basis of Freedman' s formulation for wave scattering and by applications of the Lipchitz transform under Holder conditions in fractals, a demonstration was given that the Hausdorff dimension of the solid-angle discontinuity on the scattering surface is the same as the one of the surface itself, and an expression of the scattering strength of the fractal surface has been given. A comparison with the Schulkin-Shaffer empirical formula for the sound scattering from sea surface proposes that, in this situation, the generalized (continuous) Koch surface seems to degenerate into the (discrete) four-two Cantor sets, only the latter make a contribution to the backscattering.
A surface diffusion coefficient is needed in room acoustics to enable the quality of diffusing surfaces to be evaluated. It may also facilitate more accurate geometric room acoustic models. This paper concentrates on diffusion coefficients derived from free-field polar responses. An extensive set of two- and three-dimensional measurements and predictions was used to test the worth of different diffusion coefficient definitions. The merits and problems associated with these types of coefficients are discussed, and past parameters reviewed. Two new coefficients are described. The new measure based on the autocorrelation function is forwarded as the best free-field coefficient. The strengths and weaknesses of the coefficient are defined.
Number theoretic diffusors, as described by Schroeder, are characterized by a periodic grouping of a series of wells of equal width, but different depths, separated by thin dividers. Over the past 7 years the quadratic residue depth sequence has been used in over 1000 applications, including recording studios, concert halls, worship spaces, conference rooms, music education, and even home theater and music listening rooms. The frequency bandwidth is determined principally by the depth of the deepest well, which determines the low-frequency limit, and the well width, which determines the upper frequency limit. Physical manufacturing constraints pose a space limitation of approximately 1 in (25.4 mm) on the well width and a depth limitation of approximately 16 in (406.4 mm), after which the units become diaphragmatic and defeat the purpose of increasing the depth further. In an effort to provide full-spectrum sound diffusion in a single integrated diffusor, the self-similarity property of fractals was combined with the uniform scattering property of the number theoretic reflection phase grating to produce a patented new diffusing fractal called the Diffractal. The Diffractal is in effect a diffusor within a diffusor, with each covering a different frequency range, much like a multiway loudspeaker. The mid-high-frequency diffusors are placed on the wells of a dedicated stiff and massive low-frequency diffusor. The passband of the nested band-limited diffusors and the crossover points are completely calculable, and the overall bandwidth is limited only by the available depth and the surface detail. The theory of the one- and two-dimensional diffusor, fractal geometry, and the one- and two-dimensional Diffractal is reviewed. Two installations at Real World Studios, Bath, UK, and Crawford Post, Altanta, GA, USA, are presented.
An optimization method has been used to design unbaffled and baffled curved diffusers for performance spaces. The design method requires single parameters to characterize the quality of diffusion produced. A parameter based on the standard deviation function has been refined and used for the unbaffled surfaces. For the baffled diffusers a new method was used which is based on the difference in scattering over the specular zone when the diffuser is added to the baffle. The unbaffled new diffusers match or outperform standard curved based on the arc of a circle. In some cases the arc of a circle is the best solution available. This occurs when a near semicircle is within the allowed geometry. Tests also examined forming unbaffled modular diffusers from many periods of identical curved diffusers. Similar results regarding semicircle shapes were found. Signs of lobing for many periods of the curved diffusers were seen. For baffled diffusers the new optimized surfaces always had a better performance than the arc of a circle, reducing the scattered pressure within the specular zone by 4-dB on average. Measurements confirmed that better performance was achieved for an optimized baffled surface, compared to the standard arc of a circle.
The sound field scattered by a fractal surface in the form of a Sierpinski carpet is calculated in the framework of the perturbation method. The Sierpinski carpet has an alternating acoustic admittance preset at its squares, which sequentially scale down. It is demonstrated that such a Sierpinski carpet scatters sound almost uniformly in all directions.
A ‘‘separation of scales hypothesis’’ is an important component of both composite roughness and facet scattering theories applied to fractal (i.e., realistic) surfaces. It asserts that a scattering surface is separated by the acoustic wavelength into two scale regimes; features larger than the acoustically defined ‘‘separation scale’’ act to reflect rays specularly, whereas features smaller than the separation scale act to scatter acoustic energy diffusely. This paper explores the consequences of this hypothesis by formulating the relationship between an arbitrary local‐scale scattering strength function, describing the diffuse component of scattering relative to a local‐scale slope, and three global‐scale scattering strength functions, which describe scattering over the entire surface relative to the horizontal. These include: mean and rms scattering strength, and grain ratio scattering strength (the ratio of scattering strength from two principal directions of an anisotropic surface). Each provides an important independent constraint on the relationship between local‐ and global‐scale scattering properties: (1) Mean global‐scale scattering strength is essentially a more diffuse version of local‐scale scattering strength; (2) rms global‐scale scattering strength exhibits great sensitivity at low grazing angles to the local‐scale scattering strength model; and (3) grain ratio scattering strength is sensitive to the functional form of local‐scale scattering strength, especially at low grazing angles. Examples demonstrate that sinusoidal local‐scale scattering produces sinusoidal global‐scale scattering, whereas Lambertian local‐scale scattering does not result in Lambertian global‐scale scattering.
The accuracies and computational times of theoretical methods used to predict the scattering from rigid plane and curved reflectors have been evaluated. These methods were based on different solutions of the Helmholtz-Kirchhoff integral equation. The prediction results were compared to measurements. It was found that a boundary integral method provided accurate predictions for both panel types. For the more approximate prediction methods their limitations were defined in terms of the accuracy achieved and the range over which the methods were applicable. These methods were also used to investigate the use of a cut-off frequency to describe the limit above which specular type reflections dominate the scattering. It was found to be applicable only close to the geometric scattering angle.
In room acoustics, the scattering coefficient of a diffusing surface is defined as the ratio of the non-specularly reflected power to the total power reflected by the surface. The value of this coefficient (and its dependence on frequency) is essential for the users of any modern room acoustics calculation tool. Some techniques have been proposed to measure this value, under well-defined sound field conditions. The scattering coefficient is here rather derived from scattering theories. In particular, random rough surfaces are analysed using Kirchhoff Approximation methods, which lead to theoretical expressions illustrating the dependence of the scattering coefficient on frequency, angle of incidence and geometrical roughness profile of the surface. It is shown that these theoretical expressions are consistent with the results that could be expected from the determination of the whole directional distribution of the scattered pressure.