In the preceding paper the present authors analyzed the sound field in an axially symmetric duct by regarding the duct as a continuum of acoustic transformer elements. By using the same theoretical treatment as in the preceding paper, propagation of sound in a curved duct is considered in the present paper. Simultaneous ordinary differential equations defining the sound field in a curved duct is derived and presented in Equation(11). The equations thus obtained are discussed in order to clarify their physical meaning. The practical applications of these equations, based on Fourier analysis, are also presented to show their usefulness. And we calculated the sound field in the curved ducts shown in Figs. 2 and 3 where a simple plane(0, 0)mode, exp(-iks), is incident upon them. The results are given in Chapters 4 and 5 respectively.

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The liquid-tube transducer, a relatively new form of underwater acoustic axial-radiator, is investigated. The liquid-tube transducer consists of a liquid-filled elastic tube excited at one end by a piston source and terminated at the other end by a metal plug. In case the sound velocity of the inner liquid (c_0) is smaller than that of the outer liquid (c), the normal mode--usually the (0, 1) mode--propagates along the inner liquid cylinder with a little radiation loss to the surrounding medium. Such a system actually constitutes an axial-radiator analogous to the dielectric-rod antenna. For the normal modes with a radial symmetry we have Eqs. (1) and (2). After some calculations, the characteristic equation which determines the phase velocities of the(0, n)modes is finally expressed as Eq. (15). The characteristic values and corresponding phase velocities for the first few modes are plotted in Fig. 3 as functions of frequency parameter M(Eq. (13)). The directional characteristics are obtained by computing the field produced by the equivalent density of sound source (or sound function) g[1/sec] within the tube. This equivalent density of sound source is determined by use of Kontorovich's second equivalence principle (see Eqs. (19), (20) and (21)). In terms of the equivalent density of sound source g, the velocity potential φ at a great distance is given by Eq. (22). Eq. (26) represents the field pattern produced by a disk of unit thickness and equal diameter with the tube, while Eq. (27) represents the effect of the length of the tube. Normalizing Eqs. (26) and (27), the directivity functions are expressed in the forms of Eqs. (28) and (29), where a=radius of tube, L=length of tube, x=characteristic value (cf. Fig. 3 or Fig. 8), k=ω/c, ω=circular frequency, k_z=propagation constant in z-direction (cf. Fig3 or Fig. 8), α=attenuation constant which should be determined by experimental data (cf. Fig. 7). Fig. 6 shows a section of the liquid-tube transducer. The experimental directional patterns are obtained for four tubes of various lengths between 17-52cm with common inside diameter (36mm). In Fig. 9, the broken curves represent the experimental directional patterns, corresponding to 20kHz and for three tubes of various lengths, while for the purpose of comparison the directional patterns calculated from Eqs. (28) and (29) are plotted by the solid curves. The theoretical values of k_z/k=1. 10, x=1. 54 and the experimental value of α=0. 1dB/cm at 20kHz (cf. Figs. 7 and 8) are used in obtaining the theoretical directivity in Fig. 9. Fig. 10 shows the 3dB-beam-width of the main-lobe and the first side-lobe-level as functions of the tube length at 20kHz. For a higher frequency (24kHz), the directional patterns become more complex (cf. Fig. 11) because of the larger value of k_z=1. 21. In Figs. 9 and 11, the comparatively good agreements between the theoretical and experimental curves are found for most of the tubes investigated, except the shortest one for 20kHz. In addition, it is evident that even for smaller k_z/k=1. 01 (cf. Fig. 12) or larger k_z/k=1. 33 (cf. Fig. 13) the measured curves agree considerably with those calculated. The optimum length of the tube (L_0) is determined from Eq. (31) in accordance with the Hansen-Woodyard's condition (see Fig. 14). This condition also determines the optimum value of k_z and c_z.

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