The waveguiding of sound using lines of resonant holes

The dispersion of an acoustic surface wave supported by a line of regularly spaced, open ended holes in an acrylic plate, is characterised by precise measurement of its localised acoustic fields. We illustrate the robust character of this surface wave and show its potential for control of sound by the acoustic waveguiding provided by a ring of regularly spaced holes. A single line of open-ended holes is shown to act as simple acoustic waveguide that can be readily manipulated to control the flow of sound.

www.nature.com/scientificreports www.nature.com/scientificreports/ inside one of the holes at the positions indicated in Fig. 1. This loud speaker was driven by a Gaussian-shaped electrical pulse containing a broad range of frequencies (~4-18 kHz), thereby exciting the ALM over a frequency band which includes the expected resonant frequency of each hole. The detector 'needle' microphone, mounted on a motorized translation stage and with its 0.5 mm radius tip placed less than 1 mm above the sample surface (which lies in the x-y plane), then records the pulse for a square array of points parallel to the x-y plane covering the area of the sample. This time domain data was then Fourier analysed to provide amplitude and phase for each frequency at each point in space. The dispersion of the surface wave may then be achieved by subsequent spatial Fourier analysis of each individual frequency map.
Experimental pressure-field maps illustrating excitation and propagation of the ALM on a single line of holes are presented in Fig. 2. The geometry of the open ended holes are clearly evident in the pressure fields, even at 11 kHz despite the radius r L being ~10% of the excitation wavelength of the radiation. By taking the 2D Fourier transform of the pressure field at each frequency component of the excitation pulse, the amplitude of each wave-vector-component (k x and k y ) is calculated. Then 'stacking' in frequency each of these 2D arrays of data and taking a plane through k y = 0 allows visualization of the f-k x dispersion relation of the modes supported. This procedure gives Fig. 3, the experimentally determined dispersion of the ALM supported by the line sample along the k x direction. Overlaid (green circles) are the Eigenfrequencies of the structure calculated using a finite-element-method (FEM) model. The sound line, k 0 = 2π/λ 0 , i.e. the dispersion of a plane wave in free space propagating along the surface of the sample, is shown by the white solid line (Acoustic properties for air taken from Cramer 25 ).
The ALM is visible in Fig. 3 as a strong feature in the non-radiative regime k x > k 0 , showing that it is indeed trapped and does not radiate from the surface. The mode exists with increasing wave-vector component along the surface until the first Brillouin zone boundary, where it becomes a standing wave with zero group velocity and can no longer be detected with the time-gated pulse technique. (Over most of the frequency range it has a 1/e amplitude decay length of order 10 to 200 mm.) There is a fainter curve of identical shape in the negative half of www.nature.com/scientificreports www.nature.com/scientificreports/ k-space, i.e. corresponding to a surface wave propagating in the negative x-direction. Because the excitation probe was placed near one end of the sample this dispersion arises from waves reflected from the far end of the sample.
As mentioned, the ALM arises through diffractive near-field coupling between fields in adjacent holes. Given that the fields in the holes are sufficiently well coupled, which is dependent on the frequency, geometry and spacing of the resonators, then ALMs will exist, opening up the potential for bespoke control of acoustic surface energy trapped on a surface. The second sample studied in the work, designed to illustrate this, is comprised of a ring of equally spaced holes with centre points on a ring of radius R R (Fig. 1B). Figure 4A is the experimentally determined instantaneous pressure field of the ring sample (R) excited by an acoustic pulse, and the data extracted for 14.625 kHz, corresponding to λ gR /λ 0 = 0.341. Similarly to the data from the line (L) sample, (Fig. 2.), it is clear that an ALM is supported by the structure. Following the same procedure described above to produce the data illustrated in Fig. 3, we derive the dispersion of the mode supported by the R-sample (Fig. 4B). The key difference is that the spatial Fourier transforms are performed with an orthogonal grid of polar coordinates rather than Cartesian, allowing representation of the reciprocal lattice as a function of k θ (which is proportional to the angular momentum number 24 ) and k r . Any plane of this data set taken through k r = 0 illustrates the dispersion. The change of co-ordinates also has important implications for the definition of a trapped surface wave. The angular wavelength λ θ is determined by θ × r, hence the definition of free space wavevector k 0θ = 2π/λ θ0 is not fixed. This means that at a given frequency, the speed of the curved wavefronts will depend on the radial coordinate r. At some radius, to keep the phase fronts radial, the wavefront would need to travel faster than the speed of sound c, hence the wave will always have a radiative component. To make a  . Experimental dispersion diagram for the line sample, obtained from the spatial Fourier transforms of the measured pressure fields on its surface. The magnitude of the Fourier transform is plotted as a function of the ratio of grating periodicity λ gL to incident wavelength λ 0 vs the normalized in-plane wavevector, k x / k gL , along the array surface. The first Brillouin zone is at k x /k gL = 0.5. The horizontal dot-dashed line marks the frequency to which Fig. 2 corresponds. The solid white lines represent the 'sound lines' , the maximum wavevector k 0 = 2p/l 0 that a grazing incidence sound wave can possess. The numerically calculated dispersion is represented by the overlaid green circles.
www.nature.com/scientificreports www.nature.com/scientificreports/ comparison between the dispersion of the line sample and the ring sample requires that for the ring sample we chose a value for k 0 , using an arbitrary radius to define λ θ . To maintain consistency, the radius chosen was R R to the center of each hole cavity; at this radius, the periodicity in θ (λ gR ) is equivalent to the periodicity in x (λ gL ) of the line sample (≈8 mm). This has been done in Fig. 4B, where the marked sound lines and Brillouin zones use k gR and k 0θ defined accordingly, as well as the grating periodicity λ gR .
Here again the ALM is clear in the Fourier transformed data in Fig. 4B with a a dominant feature visible in the regimes|k θ |>|k 0θ |having greater momentum than is possible for a grazing wave, at the radius R R . With increasing k θ , the mode forms a standing wave in θ at its equivalent Brillouin zone asymptote k θ /k gR = 0.5, with its decay length along θ decreasing as this asymptote is approached. The overlaid green circles represent the Eigenfrequency solutions from the FEM model. Thus, these ALMs are robust and can indeed follow lines of holes on a surface, provided the local in-plane curvature of the line is not so severe (i.e. small R R ) that the radiative losses become significant.
In conclusion, it is shown that a line of holes acts as a simple acoustic waveguide through the excitation of a trapped acoustic surface wave demonstrated via a high resolution acoustic imaging technique. Using spatial Fourier transforms, the dispersion of the mode supported by these simple hole structures are directly recorded. A single, one-dimensional line of open-ended rigid-walled holes is characterized, where it is found that the fields in the holes couple and support a strong surface wave. Secondly a line of holes is configured into the shape of a ring, where it is shown that waveguiding persists as a strong feature following the ring's circumference. This type of trapped surface wave is robust to slow arbitrary changes in direction and offers opportunity as a novel method for controlling sound. 625 kHz (l gR /l 0 = 0.341), measured just above the ring sample's surface. The point-like source was located inside the hole at r ~ 100 mm, θ = 2.3 rad. (B) Experimental dispersion diagram for the ring sample, obtained from the spatial Fourier transforms (in polar coordinates) of the measured pressure fields. The magnitude of the Fourier transform is plotted as a function of the ratio of grating periodicity λ gR to incident wavelength λ 0 vs normalized in-plane wavevector k θ , along the array surface in the θ direction, along the holes. λ gR is defined at the radius R R , the center of each hole. The solid white lines represent the 'sound lines' , the maximum wavevector k 0θ = 2p/l q0 that a grazing incidence sound wave can possess, here in the k θ direction, while the dashed white lines are the diffracted sound lines ± (k 0θ + k gR ), where k 0θ and k gR have also been defined at the radius R R. The horizontal dot-dashed line marks the frequency to which Fig. 4 corresponds. The numerically calculated dispersion is represented by the overlaid green circles.