The nonlinear interaction of sound waves in air has been applied to sound reproduction for audio applications. A directional audible sound can be generated by amplitude-modulating the ultrasound carrier with an audio signal, then transmitting it from a parametric loudspeaker. This brings the need of a computationally efficient model to describe the propagation of finite-amplitude sound beams for the system design and optimization. A quasilinear analytical solution capable of fast numerical evaluation is presented for the second-order fields of the sum-, differencefrequency and second harmonic components. It is based on a virtual-complex-source approach, wherein the source field is treated as an aggregation of a set of complex virtual sources located in complex distance, then the corresponding fundamental sound field is reduced to the computation of sums of simple functions by exploiting the integrability of Gaussian functions. By this result, the five-dimensional integral expressions for the second-order sound fields are simplified to one-dimensional integrals. Furthermore, a substantial analytical reduction to sums of single integrals also is derived for an arbitrary source distribution when the basis functions are expressible as a sum of products of trigonometric functions. The validity of the proposed method is confirmed by a comparison of numerical results with experimental data previously published for the rectangular ultrasonic transducer.
We investigate the gauge coupling renormalization in orbifold field theory preserving 4-dimensional N = 1 supersymmetry in the framework of 4-dimensional effective supergravity. As a concrete example, we consider the 5-dimensional Super-Yang-Mills theory on a slice of AdS 5 . In our approach, one-loop gauge couplings can be determined by the loop-induced axion couplings and the tree level properties of 4-dimensional effective supergravity which are much easier to be computed.Recently higher-dimensional field theories compactified on orbifold have been proposed as models providing an efficient mechanism for symmetry breaking, e.g. the supersymmetry (SUSY) breaking [1, 2] and/or the grand unified gauge symmetry breaking [3]. One can construct realistic grand unified models more efficiently in such framework [4,5]. Higherdimensional orbifold models may also lead to a geometric understanding of various hierarchical mass scales in particle physics [6,7], the suppression of some Yukawa couplings [8], and the b-t mass ratio [9].In this paper, we wish to discuss the gauge coupling renormalization in orbifold field theories preserving the 4-dimensional (4D) N = 1 supersymmetry in the framework of 4D effective supergravity (SUGRA). We are particularly interested in theories with large scale hierarchies, for instance a model in which the Kaluza-Klein (KK) threshold scale is significantly lower than the cutoff scale of the theory so that the KK towers are important for the gauge coupling renormalization. In such cases, it is quite convenient to consider the gauge coupling renormalization in the framework of 4D effective SUGRA since the 1-loop gauge couplings can be determined by the loop-induced axion couplings and the tree level properties of 4D effective SUGRA which are much easier to be computed. As a concrete example, we will consider 5D SUGRA-coupled super-Yang-Mills (SYM) theory on a slice of 5D Anti-de Sitter space (AdS 5 ) [10]- [14] with four well-seperated mass scales: the KK scale, the orbifold length, the AdS curvature, and finally the cutoff scale. However, much of the discussions here can be easily extended to generic higher dimensional orbifold field theories. Also the results for a flat supersymmetric 5D geometry can be obtained from our AdS results by taking the limit that the AdS curvature becomes zero.The organization of this paper is as follows. In section 2, we discuss some features of the supersymmetric gauge theory on a slice of AdS 5 , including the gauged U(1) R symmetry and also the possible form of gauge coupling renormalization. In section 3, we derive the
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