同軸円筒状空胴の四端子定数

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Usually, acoustical characteristics of cylindrical cavities are calculated using the onedimensional model in which are present only plane-waves travelling in the axial direction. However, the results of calculations are not always in good agreement with the results of measurements even in the range below the transverse resonant frequency. In this paper, the four-terminal constants of the co-axial cylindrical cavity are deduced by means of the velocity-pontential in the cavity in which the entrance and the exit are replaced by pistons. The velocity-potential Φ in a cavity of radius a and length l, illustrated in Fig. 1, is given by the summation of Φ_1 and Φ_2, where Φ_1 is the velocity-pontential when the exit piston is fixed and Φ_2 is the one when the entrance piston is fixed, these are given by equations (2) and (4). When the cavity is not extremely flat, and when ka is smaller than λ_1 and is not too near λ_1, where ka=λ_1 is the first transverse resonance, the sound pressure P_i(r, z_0) on the end-plate of the vibrating piston and P_i(r, z_1) on the end-plate of the fixed piston, the average sound pressure P^^^-_<ii> on the vibrating piston and P^^^-_<ii> on the fixed piston are given by equations (13), (14) and (15), respectively, where i=1 or 2 corresponding to Φ_1 or Φ_2 and i′=1 or 2 (i′≠i). An example of P_i(r, z_0) and P^^^-_<ii> is shown in Fig. 2, where P′_i(r, z_0) is the sound pressure of the one-dimensional model [eq. (16)]. Fig. 3 shows some results of calculations of G_i using eq. (12), and when ka is smaller than about 1. 5, G_i is found to have the form given by eq. (17). Fig. 4 shows g_i of eq. (17), and when a_i/a is smaller than 0. 4, g_i is found to be given by eq. (18). Since total sound pressures are given by eq. (19) and the relation of the volumevelocities is given by eq. (21) when the exit impedance is equal to zero, the four-terminal constants lead to eq. (23), and when ka<λ_1 these constants are given by eq. (25). If G_i=0, eq. (25) gives the constants of a one-dimensional model. The results of calculations are shown in Figs. 5 and 6. Fig. 7 shows the constant A obtained by various methods. Though the boundary conditions are somewhat different from each other, the results seem to be nearly equal. Total four-terminal constants of the network illustrated in Fig. 8 are given by eq. (26), where A′, B′, C′ and D′ are the four-terminal constants of the one-dimensional model and X_1 or X_2 is the reactance. By comparing eq. (25) with eq. (26), in the range below the transverse resonance, it is assumed that the co-axial cylindrical cavity is equivalent to a one-dimensional model of same dimensions which is connected by reactances X_i[eq. (27)] at each end. When ka<1. 5, X_i is equal to the reactance of a pipe of length ΔZ_i=a_ig_i connected to the entrance or to the exit. As a result, the four-terminal constants or other characteristics of co-axial cylindrical cavities cannot be calculated using only a one-dimensional model, but require a model with end reactances. In the appendix, it is shown that the exact expression of the "transmission loss" TL for cylindrical cavities is given by eq. (33), and the "noise reduction" NR is given by eq. (34) which is nearly equal to the 20 log |A|. However, the author emphasizes TL and NR cannot show the effectiveness of mufflers.
社団法人日本音響学会の論文
1975-11-01

著者

奥田襄介
熊本大学工学部

同軸円筒状空胴の四端子定数

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