# Surface Impedance Boundary Conditions: A Compre...

However, these approaches modulate radiation patterns by adjusting the material properties throughout the propagation region. However, for widespread situations, it is impossible to do so. In addition to adjusting the material properties, adjusting the surface impedance may be another effective way to modulate radiation patterns. In electromagnetism, a smooth conducting sheet has low surface impedance. A periodically grooved subwavelength structure in the surface of metal can convert low surface impedance to high surface impedance18. However, in acoustics, we would like to convert high acoustic surface impedance to low acoustic surface impedance. Another property of metals is that they support surface waves19, but acoustic surface waves do not exist in fluid domain.

## Surface Impedance Boundary Conditions: A Compre...

In this study, in the acoustic scope, we propose a method that utilizes boundaries instead of an entire propagation region to control the radiation pattern, which will greatly reduce the complexity in pattern control and expand the application of wave modulation. Compared with utilizing the interference to control the radiation pattern and realizing waves such as cosine-Bessel beams, utilizing boundaries to control the radiation pattern will greatly reduce the scale of the source. The realization of Bessel beam is through the superposition of an infinite number of plane waves that need a source with large scale4; however, in our proposal, we can control the radiation pattern by utilizing appropriate boundaries with a point source. A system consisting of a single slit surrounded by finite, periodically drilled cavities is designed and fabricated. In this structured system, both the experiment and the numerical calculation demonstrate that the dipole-like radiation pattern can occur over a relatively wide frequency range. It may seem that our system is similar to that of a single slit surrounded by finite, periodically perforated grooves; however, the experimental phenomenon and the mechanism of our system are quite different. The directional beam experimental report by Zhou et al.20 showed very strong fingerprints of the running surface wave scattered by the grooves20,21,22, and the directional beam occurred in a very narrow bandwidth. We verify that the dipole-like pattern radiation phenomenon in our samples arises from the effective boundary impedance adjustment. The periodically drilled cavity is designed to adjust the effective impedance of the boundary, creating a dipole-like radiation pattern with no sidelobes.

It is known that if a point source is put on an infinity boundary with either a rigid or free boundary, the acoustic field can be regarded as that produced by the source and an imaginary source with the same phase or with an opposing phase in the symmetric position. If the point source is near the boundary, the source and the imaginary source can be regarded as either a monopole or a dipole24,25. It seems that the undecorated sample will induce the monopole-like fields. For the decorated sample, if the radiation pattern can be attributed to the impedance adjustment by the HRs, thus changing the rigid boundary to be free or quasi-free, the dipole or dipole-like radiation field caused by the decorated sample can be understood.

How to cite this article: Quan, L. et al. Effective impedance boundary optimization and its contribution to dipole radiation and radiation pattern control. Nat. Commun. 5:3188 doi: 10.1038/ncomms4188 (2014).

Sound field in a rectangular enclosed space is an important and classical problem in acoustics and has applications to office buildings, production facilities, cars, ship cabins, and reverberant rooms, among others. Understanding this problem helps with predicting sound fields and controlling noise. In applications, it is usually more practical to consider a rectangular enclosed space with general impedance boundary conditions than one with ideal boundaries, but the former approach is yet to be studied fully in the field of acoustics. A nonanechoic tank is an important acoustic measuring device that is used widely for calibrating transducers [1, 2] and measuring sound power [3, 4] and the sound absorption coefficient of underwater acoustic materials [5, 6]. Being able to predict the sound field in a nonanechoic tank reasonably and effectively will help understand the sound field characteristics and provide the necessary basis for acoustic measurements using nonanechoic tanks.

The accuracy of predicting the sound field in a nonanechoic tank depends largely on the correctness with which the boundaries of the sound field are modeled. Compared with an ideal boundary, an impedance boundary is more in line with an actual sound field. An impedance boundary combines both the attenuation of the energy amplitude and the phase change of the response, and it has obvious advantages compared with an absorption boundary [7].

Classical numerical methods for solving nonlinear equations (e.g., Newton iteration) are not suitable for solving the aforementioned equations because they yield only one root for a single initial guess. Naka et al. [10] proposed an interval Newton/generalized bisection method to help solve such nonlinear equations. Bistafa and Morrissey [11] proposed a new numerical method in which the eigenvalue problem is posed as one of homotopic continuation from a nonphysical reference configuration in which all eigenvalues are known and obvious. Du et al. [12] proposed using Fourier series to analyze the acoustic field in a rectangular cavity with general impedance boundary conditions, thereby transforming the problem of finding the characteristic root of a nonlinear transcendental equation into that of finding a standard matrix eigenvalue. However, the aforementioned approaches require the sound field to be rectangular. Instead, Xie et al. [13] proposed an approach based on a weak variational principle to study the sound field inside an acoustic enclosure whose walls have arbitrary inclinations and impedance conditions.

Finite element method is one of the widely used numerical analysis techniques. Because it can be applied to almost all the continuous medium problems or field problems, the finite element method has attracted great attention in various fields of physics including acoustics. Thanks to the advances in computer technology, the acoustic finite element method is nowadays widely used to solve the acoustic problems. Vorlander [16] investigated the concepts and uncertainties of computer simulations in room acoustics, and he believes that the reliability depends on the skills of choosing the correct input data of boundary conditions such as absorption and scattering. Aretz and Vorlander [17] investigates the influence of different boundary representations of porous absorbers on the simulated sound field in small rooms. They analyzed the sound field of a scale model room with a well-defined geometry through experimental measurements and finite element methods. Naka et al. [18] combined the geometric methods and finite element method to calculate the sound field in rooms with realistic impedance boundary conditions.

All the methods mentioned above require the room to have a regular shape. In the present paper, a method is presented for using acoustic software to calculate an impedance-boundary sound field. The advantages of this method are that it is not limited by the shape of the sound field during the calculation and the impedance value can be arbitrary. There is no specific requirement for the material of the tank wall. As long as the parameters of the material are known, the sound field prediction can be performed using this method. This method uses the principle of a standing-wave tube (SWT) to measure the acoustic impedance of materials, and a virtual SWT is established in software to obtain the impedance value at the boundary of the sound field. With the obtained impedance value, the sound field in a water tank with the same impedance boundaries can be calculated. We use the method proposed herein to calculate the sound field due to a point source in a glass water tank. By comparing the numerical results with experimental results, we verify the accuracy of the method. We then calculate and compare the sound fields in a large nonanechoic pool with different impedance values and analyze how the boundary impedance values influence the sound field.

A rectangular enclosed space of dimensions and the associated coordinate system are shown in Figure 1. For an enclosed space with uniform impedance on each of its walls, the Helmholtz equation can be written aswhere is the driving wave number with frequency and is the speed of sound. The Helmholtz equation can be solved by separating the variables. Because the solutions in different directions have the same form, we take the x direction as the example here. The eigenfunction in the x direction can be written aswhere is the wave number in the x direction and A and B are constants. The boundary conditions in the x direction can be written aswhere is the outward normal unit vector on the walls, is the density of the medium, and is the impedance of the walls. Substituting equation (2) into equation (3) gives

Since equation (5) is nonlinear, there is no way to analyze the sound field problem in the enclosed space of the impedance boundary by analytical calculation. This problem can only be solved by numerical calculation.

Although it is complicated to calculate a three-dimensional sound field with impedance boundaries, the sound field in an SWT with an impedance boundary at the end of the tube is easy to calculate. To verify the ability of the finite-element (FE) software Actran and to calculate a sound field with impedance boundaries, the sound field in an SWT with an impedance boundary at the end of the tube is calculated by the FE method and analytically.

As shown in Figure 4, the normalized results of the analytical and numerical calculations are consistent. Thus, the accuracy of using the FE software Actran to calculate a sound field with an impedance boundary is verified. 041b061a72