Various kinds of modes will be excited and modulated in an acoustic transmission duct due to variations of its geometrical conditions. Therefore, we may be able to consider an acoustic transmission duct as a continuum of acoustic transformer elements. In the first part of this paper, a well-known one-dimensional wave equation is derived from the above point of view. Then more accurate differential equations applicable to the analysis of an axially symmetric transmission duct are derived with higher order modes taken into consideration and are presented in Equation (19). The equations thus obtained are discussed in order to clarify their physical meaning. The theoretical consideration treated here may be applied to another type of acoustic transmission duct which will be presented in the next paper.
The linear array or the curved one is useful to the sound source of sound reinforcement system, and these arrays are utilized in many applications. In order to obtain desirable directive pattern of the array, the curved array is superior to linear one. However, it is quite difficult to determine the optimum arrangement of the curved array theoretically. In this paper, a method to design the curved array is proposed. It is shown theoretically and experimentally that adequate sound pressure distribution can be obtained by the curved array designed by the proposed method. The shape of the curved array is determined by the following method. That is, the region of audiences' seats in the sectional plan of the room is divided into equal M parts as shown in Fig. 1-(b) and equidistance points from the rear wall are denoted by P_1, P_2. . . , and P_<M+1>. By the use of the angles (θ_n, n=1, 2, . . . , M) between the straight line S_0P_n(n=1, 2, . . . , M) and the horizontal line, the arrangement of loudspeakers is determined as shown in Fig. 1-(a). The results of numerical computation on the near field sound pressure distribution of the curved array synthesized by the above-mentioned method are shown in Figs. 4, 5, 6 and 8. In comparison between the near field sound pressure distribution of the curved array (Figs. 4, 5, 6 and 8) and that of the linear array (Fig. 9) it is clear in Figs. 4, 5, 6, 8 and 9 that the sound pressure distribution of the curved array within the objective region is flatter than that of the linear one. Furthermore, the changes of the sound pressure distribution with respect to frequency in the case of the curved array are less than those of the linear array. Experiments on the sound pressure distributions of the curved array were carried out in order to verify the theoretical considerations. It is observed in the experiments that there exists some difference between the results of the curved array and the linear one, as shown in Figs. 11, 12 and 13.