Excellent theoretical works have been accomplished over the past two decades on the scattering of acoustic waves by randomly rough surfaces. In most of those works the starting point has been the approximation of surface boundary conditions by the physical-optical approximations, except in the cases of Wagner and Lynch. The former analyzed the shadowing phenomena of randomly rough surfaces and the latter considered improving the surface field expression based on the physical-optical approximation by making curvature corrections. A basic assumption for irregular surfaces which is common to all the work is that the roughness, i. e. , deviation from the mean level is given by a single Gaussian distribution. On the other hand Beckmann investigated the scattering of electromagnetic waves by a composite rough surface whose roughness is expressed as the sum of several independent Gaussian random functions. This paper is an extension to the case of the scattering of acoustic waves by a composite rough surface whose height function is given as the sum of a finite number of statistically dependent Gaussian random variables [Eq. (31)]. The sea-bottom is made up of the basic profile (continental shelf) and the small-structure roughnesses caused primarily by rocks, coral reef, pebbles, sands, mud, sea weeds, etc. , and the natural evidence about the sea-bottom indicates that the small-structure roughnesses are definitely dependent upon the basic profile, i. e. , the large-structure roughness. In view of this the statistical model of rough surfaces proposed here is thought of as more natural and better than the ones presented by Beckmann and others. A coordinate system for describing the scattered field is shown in Fig. 1, and Fig. 2 explains the geometry of the scattering. It is assumed that the wave equation (7) holds throughout the medium bounded by the rough surface and the sphere at infinity, and Green's theorem provides Eq. (8) for the reflected pressure φ_r(P) at an observation point P. For simplicity the boundary condition at the surface is taken to be zero pressure - the "free surface" condition. Following Beckmann's approach the mean intensity of the scattered pressure wave is obtained as Eq. (32), where ζ is the height function given by Eq. (31). ζ_p(p=1, 2, . . . . , n) are Gaussian random variables with zero means and variances σ_p^2. It is also assumed that ζ_p is correlated with ζ_q by the correlation function ρ_<pq>. The characteristic function in the integrand of Eq. (32) can be expanded as Eq. (36)(see Rice), where C_<pp>(τ), C_<pq>(τ) are correlation coefficients given in Eqs. (38). The particular forms of the correlation coefficients are known to fit quite well many contour maps. For a very rough surface for which the relation (39) holds the mean intensity of the scattered pressure wave is now given as in Eq. (42). For a rough surface which may be considered to consist of a large-structure roughness and a small-structure roughness n=2, and Eq. (42) yields Eq. (44).
In the application of longitudinal vibration, a conventional vibration system is usually of one-dimensional construction, such as a solid horn. High power vibrational energy is not contained because of the radial coupling of the vibration. But however, the authors are sure it may be more convenient that ultrasonic energy can be transmitted from the direction of the driver to the other, and also, that ultrasonic energy may be concentrated by the parallel operation of drivers and may be divided into plural loads from one vibrational source. From this point of view, the authors have devised the directional converters of longitudinal vibration (called an L-L-L Type Converter) as shown in Fig. 1. The converter consists of three longitudinal vibrational robs of equivalent half wavelength coupled with each other at their node of vibration (as shown in Fig. 5), and it has two fundamental resonant frequencies. One is in the same phase at their free end surfaces (called in-phase-mode resonance (as shown in Fig. 4a)), and the other is in the same phase with each other at two directional free end surfaces and the opposite phase at the other free end surface (called anti-phase-mode resonance (as shown in Fig. 4(b)). In this case, each rod of it makes three dimensional vibrations to each mode. The uniform rod with rectangular cross-section as well as the converter has all three, in -phase-mode, quasi-in-phase-mode and anti-phase-mode (Table 2) resonant frequencies. In all cases, the resonant frequencies of the in-phase-mode were higher than that of the anti-phase-mode. The authors analyzed the converter by making assumptions from these vibrational characteristics, and by defining the equivalent elastic modulus of the coupled part to each axis(Table 3, 4, 5, and Fig. 10), and formed the equations (9)-(17) according to the necessary design. we also, designed, devised and measured the new converter. The result was within one percent of the designed resonant frequency (Fig. 11, 12), the calculated curve of vibrational distribution (Fig. 13), and the calculated ratio of velocity(Fig. 14, 15).
This paper deals with an analysis of transducer-resonator which is suitable for piezoelectric mechanical filter for low frequencies. The transducer-resonator consists of two pieces of piezoelectric ceramic with a piece of base beam sandwiched between them, and the upper and lower electrodes are connected with each other electrically (Fig. 1(a)). Accordingly, the odd modes of the transverse vibration are excited, but the even modes are not. As the central point o o' in Fig. 1(a) is regarded as the "sliding end" of the mechanical beam, the analysis of the vibration of Fig. 1(a) can be accomplished as a vibration in the "free-sliding end beam" as shown in Fig. 1(b). The treatment of mechanical beam shown in Fig. 1(b) is based on the simple bending theory normally used for analysis of uniform flexure beam. First, the vibration modes of transducer-resonator are given and the results are compared with the vibration modes obtained by the mechanical network theory. As the impedance of piezoelectric ceramics partially bonded to drive the base beam cannot be neglected in general, as compared with the impedance of the base beam, the resonant frequency and the vibration mode of the transducer-resonator differ from those of the base beam only (Fig. 5, Fig. 6). Secondly, the force factor, equivalent constants and the method of suppression of higher harmonics of the transducer-resonator are shown. The third mode is eliminated in case of the length ratio l_I/l≒0. 65 for the thickness ratio t_p/t_r=1. 2, l_I/l≒0. 6 for t_p/t_r=0. 6, and l_I/l≒0. 58 for t_p/t_r=0. 3, respectively (Fig. 7, Fig. 8). Lastly, the equivalent circuit of mechanical filter taking due consideration of higher harmonics of transducer-resonator is described, and it is shown that the transducer-resonator of the higher harmonic suppression type is advantageous to simplify the equivalent circuit and decrease the spurious response in the filter characteristics (Fig. 13).