Results of laboratory measurements of sound transmission through 5 × 10 arrays of meter long polyvinyl chloride pipes with lattice constants of 5 and 10 cm with filling fractions of 13% and 50% located either on medium density fibreboard or a layer of felt are reported. Ground effects and sonic crystal effects are found to be additive. Measurements and predictions show that, while there is little broadband advantage in a periodic configuration compared with a random one, a quasi-periodic arrangement in which the perturbation has a standard deviation equal to the scatterer diameter gives the best overall attenuation.

## I. Introduction

There is considerable interest in the acoustical properties of 2D periodic structures, known as phononic band gap materials or sonic crystals, which stop sound waves passing through them in certain frequency bands. The mid frequency, *f*_{0}, of the lowest band gap is known as the first Bragg frequency and is related to the periodicity, *a*, of a square lattice structure by *f*_{0} = *c*_{0}/2*a*, where *c*_{0} is the speed of sound in the embedding fluid. Recently, sonic crystals made from arrays of circular cylinders have been investigated as the basis for an alternative form of traffic noise barrier. The band gap frequencies depend on the periodicity and, to a lesser extent, the filling fraction (the ratio of the cross-section area of an element to the unit cell area, i.e., the scatterer size). While this allows the band gaps of sonic crystal barriers to be “tuned” for example to a traffic noise spectrum, they do not perform as barriers at frequencies below the first Bragg band gap and between band gaps, i.e., in the “pass bands.” Efforts have been made to increase the attenuation at lower frequencies by having complex or composite elements. For example, hollow tubes with a narrow slit along their axes act as split ring resonators and result in an additional attenuation peak below the first Bragg frequency.^{1} Composite elements consisting of concentric elastic shells and cylinders with symmetrical slits^{2} give rise to multiple attenuation peaks as does a Matroyushka (Russian Doll) arrangement of concentric slit cylinders.^{3}

Much previous research assumes the sonic crystal structure to be in the free field, i.e., no account has been taken of the presence of the ground surface. A conventional, wall type, barrier reduces the ground effect as a result of the increased mean source-receiver path height. Indeed ISO 9613-2^{4} recommends that the attenuation due to ground effect (the result of destructive interference between direct and ground reflected wave components) is removed from calculations of total attenuation in the presence of a barrier. On the other hand laboratory measurements and predictions for sonic crystals consisting of cylinders with their axes parallel to “hard” and “soft” ground surfaces have shown that ground effect can enhance the overall attenuation.^{5} In many sonic crystal barrier arrangements, for example those involving trees, scatterer axes will be normal to the ground surface. Swearingen has developed a semi-analytical theory of sound scattering from a single circular cylinder positioned normal to a “soft” ground surface.^{6} A numerical method such as the boundary element method is appropriate for making predictions for this 3D problem but it is computationally demanding and, as yet, there are no published results of such an application.

The potential of regularly spaced trees for noise reduction has been investigated using nursery trees.^{7} Typically trees would have to be planted unrealistically close together to offer a substantial attenuation. The finite difference time domain (FDTD) numerical method has been used to simulate the acoustical performance of tree belts including ground, tree trunks, understorey and foliage.^{8} It has been predicted that a practically realizable 15 m wide tree belt consisting of trees of at least 22 cm diameter with a lattice constant of 2 m could give an overall attenuation for passenger car noise of at least 6 dB of which 3 dB is contributed by the “soft” ground effect. It was thereby concluded that 15 m wide vegetation belts could compete with the traffic noise insertion loss of a thin, classical noise barrier (on grassland) with a height of between 1 m and 1.5 m in a non-refracting atmosphere. The effects of deliberate introduction of aperiodicity were studied. Simulations showed that “inducing some degree of randomness, either in stem centre location, tree diameter, or by omitting a number of rows (assuming that the “soft” forest floor develops in the open zones), hardly affects the predicted attenuation values.” However in Ref. 9 small deviations from regular planting schemes (pseudo-randomness) were shown to be optimal for road traffic noise shielding in case of a fixed (averaged) tree density. To be realistic in respect of tree planting schemes the study was mainly limited to filling fractions of less than 12%. Moreover, to perform the former study,^{8} a 2D FDTD method was extended to 3D by a “mirror planes” technique which restricted consideration of randomness to that only in the direction parallel to the road source.

Section II of this letter reports laboratory investigations of the effects of aperiodicity on the acoustical performance of 2D periodic arrays of identical cylinders with their axes perpendicular to a ground plane. Section III compares experimental data and predictions thereby demonstrating that (multiple) scattering and ground effects are additive. Section IV details the comparative effects of periodic, perturbed and random configurations and Sec. V gives concluding remarks.

## II. Experiments

Anechoic chamber measurements have been made using commercially available 1 m long polyvinyl chloride (PVC) pipes with outer diameter 0.04 m and wall thickness of 0.002 m in a 5 × 10 square lattice arrangement with midpoint to midpoint distance between adjacent pipes (lattice constant, *a*) equal to either 0.1 or 0.05 m corresponding to filling fractions of 13% and 50% and first Bragg frequencies of 1715 Hz and 3430 Hz, respectively. Single cylinder scattering measurements confirmed that the cylinders were acoustically hard.^{1} The pipes were placed either on a medium density fibreboard (MDF) to simulate acoustically hard ground surface or on a (1.2 cm thick) layer of felt supported by MDF to simulate an acoustically-soft surface. The sound source was a Tannoy driver fitted with a 1 m long tube, of 3 cm internal diameter. This was controlled by a maximum-length sequence (MLS) system enabling determination of impulse responses. Most of the energy of the acoustic pulse was contained between 400 Hz to 20 kHz. Outside this range, the signal to noise ratio was too low for reliable results. Care was taken to reduce unwanted reflections from the supporting rods and microphone holder by covering them with absorbent material. A free-field 1/2 in. Brüel & Kjær^{®} microphone was positioned behind the array and at 1 m distance from the source. The microphone was 0.05 m from the nearest face of the array. The distance of the source from the nearest face of the array depended on the lattice constant, being 0.51 m, for example, when the lattice constant was 0.1 m. Data are quoted in Sec. III for two source heights (0.02 m and 0.1 m) and a receiver height of 0.1 m. Figure 1 shows photographs of experimental arrangements involving periodic and aperiodic arrays.

The transmission loss (TL) for each source-receiver geometry (with respect to free field) was calculated by subtracting signals received (1) without and (2) with the cylinder array present. To obtain the acoustical characteristics of the ground surfaces, short range excess attenuation (EA) spectra were measured.^{10,11} Best fits to these data were obtained using a two-parameter impedance model^{12} with the parameter values listed in Table I.

## III. Data and predictions

Example data for the hard ground surface are shown in Figs. 2(a) and 2(b). For source height 0.02 m, receiver height 0.1 m and range 1.0 m [Fig. 2(a)], the stop band corresponding to the first Bragg frequency is visible at 1.715 kHz as expected. For this low source height, the first destructive interference between direct and ground-reflected sound over MDF occurs at a relatively high frequency so the measured attenuation is mainly due to multiple scattering in the cylinder array. When the source height is raised to 0.1 m, the first destructive interference over the MDF board occurs near 9 kHz where it adds to the overall attenuation [see Fig. 2(b)]. Similar data have been obtained over the MDF-backed felt layer and are shown in Figs. 2(c) and 2(d). As expected over this relatively “soft” surface, for a given source-receiver geometry, the first ground effect destructive interference occurs at a significantly lower frequency (near 3 kHz for the 0.02 m high source and near 2 kHz for the 0.1 m high source) than observed over the MDF.

To predict the attenuation of an array of cylinders supported by a flat surface and with their axes normal to it, first a 2D Multiple Scattering Theory^{13} has been used to calculate the sonic crystal effect in the absence of the ground plane. To these predictions in dB have been added the ground effect spectrum (the negative of the Excess Attenuation spectrum^{12}) calculated for the surface (without the array) also in dB at each frequency point. Figure 3 shows four example comparisons between predictions of the overall attenuation (with respect to free field) and data. The general agreement between predictions and data confirms that, for filling fractions between 13% and 50%, the attenuation due to sonic crystals on a ground plane can be predicted by adding the loss due to multiple scattering calculated in the absence of ground and ground effects calculated in the absence of the sonic crystal array.

## IV. Effects of aperiodicity

Two levels of random perturbation of the regular arrangement of PVC pipes with lattice constant 0.1 m have been investigated; one with standard deviation equal to 0.5*r* where *r* is the cylinder radius; and one where the standard deviation was 2.0*r.* Let *O*(*x _{j}, y_{j}*) be the coordinates of the position of

*j*th element in the periodic array. Then position of the perturbed element is given by

*O*(

*x*+

_{j}*ζ*) where

_{j}, y_{j}+ξ_{j}*ζ*and

_{j}*ξ*are a pair of numbers drawn from a random number set with a normal distribution and a standard deviation equal to 0.5

_{j}*r*or 2.0

*r*. Figures 3(c) and 3(d) compare measured attenuation spectra with predictions obtained by adding predicted multiple scattering and ground effects as described earlier. Figures 4(a) and 4(b) compare attenuation data for the larger perturbation with those for a regular array for two source-receiver geometries.

Measurements and calculations have been made also for a totally random array of 50 PVC pipes. The periodic array of 5 × 10 elements occupy a rectangular area (9*a* + 2*r*) × (4*a*+2*r*). Locations of cylinders were determined by generating two sets of random numbers within the intervals [0, (9*a*+2*r*)] and [0, (4*a*+2*r*)] for *x-* and *y-*coordinates, respectively. Thus, the mean separation of cylinders was the same as the lattice constant of the periodic array. Figures 4(c) and 4(d) compare measured attenuation spectra for all array configurations including periodic, perturbed and random over MDF board. For arrays with a filling fraction of 13%, the small perturbation does not produce a significant change in measured or predicted attenuation. While there appears to be no broadband advantage in the attenuation spectra predicted or measured for a periodic system compared with those measured or predicted for a random array containing the same number of elements and having the same mean center-to-center spacing, measurements and predictions suggest that a quasi-periodic array with a perturbation in cylinder location with an S.D. of 2.0*r* performs better at high frequencies than either periodic or random arrangements while also reducing the negative attenuation associated with the pass bands (focusing).

## V. Concluding remarks

Laboratory experiments have investigated sound transmission at filling fractions of 13% and 50% through arrays of identical cylinders with their axes perpendicular to the ground. It has been demonstrated through comparisons between predictions and data, that, for the arrays studied, 2D multiple scattering effects and ground effects can be added. This confirms recent numerical simulations demonstrating that addition of 2D scattering effects calculated using FDTD and discontinuous ground effects calculated using GFPE gives the same results as a 3D FDTD calculation.^{9} The retention of ground effect is potentially exploitable and is not available with conventional “fence” barriers. It has been found that for filling fractions between 13% and 50% the deliberate introduction of perturbations in cylinder location with a standard deviation of the scatterer diameter can result in a significantly enhanced broadband transmission loss compared with periodic or random configurations with the same mean lattice constant and filling fraction. The question arises of the extent to which these findings would be true at filling fractions higher than 50%.

## Acknowledgments

The research leading to these results has received funding from the European Community's Seventh Framework Programme (FP7/2007-2013) under grant agreement no. 234306, collaborative project HOSANNA and from the Engineering and Physical Sciences Research Council (UK) through Grant number EP/E063136/1.