Sabine's theory stands on the following two assuptions : 1. Uniform, diffuse distribution of sound energy throughout a room at any instant, 2. equal probability of propagation of sound in all directions. Obviously, the Sabine's theory does not follow the wave theory of normal modes. Therefore, although the results obtained from the Sabine's theory are very useful and practical, the theoretical background seems to be very vague. Morse and Bolt assumed that the sound field developed by Sabine is based on the assumption of the ergodicity. It seems however that the correspondence of the ergodicity with the two assumptions has not been explained fully. On the other hand, the problem of room acoustics was studied statistically by many investigators, e. g. 3), 4), 5), based on the assumption of the randomness of the sound field. But, it seems also that the relation between the ergodicity and the Sabine's theory is not yet explained fully. Therefore, we discuss the relation between the Sabine's theory and the ergodicity in the stedy state. If the non-deterministic sound field is made up in a room by use of the band noise, we can regard the sound energy density at every point in the room as non-deterministic. The time average of the energy density is shown by equation (1). Introducing a constant quantity shown by equation (2) and the ergodicity, the time average equals the ensemble average given by equation (3). It shows that the assumption 1 by Sabine, that is, ''uniform, diffuse distribution of sound energy throughout a room at any instant", is satisfied by the ergodicity. But, the assumption 2 is not always satisfied by the ergodicity. For, the asumption 2 depends on the ensemble average only, as shown by equations (5, 6, 7, 8). Therefore, we can find that the Sabine's theory in the steady states stands on two assumptions, that is, the ergodicity and equal probability of propagation of sound in all directions essentially. Both assumptions are independent, so the assessment of the diffusibility must be performed taking the both assumptions into account.

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In actual sonar transducer arrays enclosed in a sonar dome to eliminate hydrodynamic noise, there are two kinds of acoustic interaction, one is the well-known interaction effect of mutual radiation impedance between the transducer elements, and the other is the interaction between sonar dome and transducer elements. If the sonar dome is considered to be a flat plate, the latter interaction can be dealt with as a problem involving mutual radiation impedance between the transducer elements and their images. In this paper, the numerical calculation method of determining the mutual radiation impedance coefficient between circular pistons and their images, as well as between two circular pistons on an infinite planar baffle is presented. Generally, the mutual radiation impedance coefficient between uniformly vibrating pistons is formulated by an expression of double surface integral (1), but it is difficult to solve it analytically except for special cases and to calculate concrete numerical values. So, in order to calculate these values easily, utilizing the axial symmetry of the radiated pressure field from a circular piston and two ring functions, this double surface integral expression is transformed into a double line integral expression (18). Then, computation of this expression is performed with a digital computer by the numerical integration method. The values of numerical mutual radiation impedance, obtained by this method for two circular pistons mounted on the planar baffle, are in good agreement with the values obtained by Klapman by analytical direct integration and by Porter by the Bouwkamp's method (Table 1). Furthermore, to illustrate the applicability of this method, numerical results are also shown in case a small circular piston is included in the large circular piston (Fig. 3 and Fig. 4), and in case the flat reflector is placed in front of two circular pistons (Fig. 5 and Fig. 6).

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This paper deals with theoretical and experimental investigations of jet noise. The theoretical analysis is made on the directivity of jet noise, the frequency of peak spectra and the modified Strouhal number, by assuming that a sound level vanishes when it propagates through the jet flow. This analysis is an extension of the supersonic jet flow theory to the subsonic flow. Experimental investigation is performed by using model jet flow. The diameter of the jet nozzle is 5 cm and the exhaust jet velocity is 300 m/sec or less. The theoretical results are in good agreement with the experimental results. Also, it can be verified that the jet noise is attenuated by diffusing the jet flow with a diffuser, and the speaker sound vanishes when it propagates through the jet flow.

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An analysis is made of nonlinear oscillations excited in a piezoelectric resonator by repeated impacts with a sinusoidally vibrating wall. The simulation method using an electrical equivalent circuit is employed for this analysis. This method gives better approximation to actual nonlinear oscillations than the conventional methods because the local deformation is taken into account. The simulation circuit for the impacting oscillatory system is shown in Fig. 2, where s_N is a virtual elastic component of nonlinear stiffness and s_h (a stiffiness of the local deformation) for the contact period (see Fig. 13). Using this circuit, various nonlinear oscillations are observed according to the frequency f of the vibrating wall (see Figs. 3, 4 and 12). The larger s_h/s'_1, the wider the frequency range in which these oscillations can be excited becomes, where 8'_1 is the equivalent stiffness of the fundamental mode. When s_n/s'_1 is less than about 0. 5, only stable one-half subharmonic oscillation can be excited within the frequency range f/f_1≧2, where f_1 is the natural frequency of the fundamental mode. A typical waveform and the amplitude characteristic of the l/2-harmonic oscillation are shown in Fig. 4(c) and Fig. 6, respectively. The influence of the zero mode and the 2nd mode are investigated on the stability and the excitation conditions of the 1/2-harmonic oscillation. As shown in Fig. 8, the 1/2-harmonic oscillation is apt to be unstable without the zero mode, and the excitation frequency range of the stable 1/2-harmonic oscillation becomes widest when m_0/m_1 is nearly unity, where m_0 and m_1 are the equivalent masses of the zero and the 2nd modes, respectively. The behavior of the stable 1/2-harmonic oscillation is not affected very much by the 2nd mode under the condition f_2/f_1≧4. 0, where f_2 is the natural frequency of the 2nd mode. The 2nd mode makes its excitation level very small when f_2/f_1 is a little larger than 2. 0 (see Fig. 11 (c)). In addition, it became clear that the zero mode is apt to made the higher harmonic oscillation unstable. The experimental demonstration of nonlinear oscillation was made. Experimental set-up is shown in Fig. 14 and the nonlinear oscillations observed are shown by the photos in Fig. 15. It was found that the qualitative behavior of a practical impact oscillatory device is as predicted by this simulation. The analysis in this paper will provide very useful data on the design of impact oscillatory devices such as a multi-mode piezoelectric frequency divider.

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A listening room is a place where a recorded sound mixed with an appropriate amount of reverberation is reproduced, where we will hear first the direct sound from a speaker and then the reflected sound from the room surface. Since the direct sound is superposed by the reflected sound, the reverberant sound of a source heard by a listener in a room changes with the increasing amount of reflection. Such change is a qualitative one that gives us an impression of acoustic spaciousness of a hall. There is another quantitative change which is expressed here by the change of the definition of a sound. The calculated definition of reproduced sound in a room depends on the reverberation time of room and the acoustic structure of recording source (See Figs. 1-3). As a result, the reverberation time advisable for a listening room is shown in Fig. 4.

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In this paper, it is investigated for a trapped-energy ceramic resonator that the value of the elastic stiffness constant c^&ltD'&gt_&lt44&gt in the plated region is lower than that of c^D_&lt44&gt in the unplated region of a piezoelectric ceramic plate in the thickness-shear mode, for the purpose of reducing unwanted responses (Fig. 1). In the construction, the elastic effect R_c similar to high plate-back ratio is expected to be obtained easily (Fig. 2). The desirable values of c^D_&lt44&gt and c^&ltD'&gt_&lt44&gt are expected to be obtained from the difference between the thoroughness of poling in the plated and unplated regions. For the proof of the expectation, it is investigated that the direction of poling in the plated region differs from that in the unplated region parallel to major surfaces of a piezoelectric ceramic plate (Fig. 3). The electromechanical coupling factor k'_&lt15&gt in the plated region decreases monotonously with increasing the angle θ (Fig. 4). The constant c^&ltD'&gt_44 decreases monotonously with decreasing k'_&lt15&gt (Fig. 5). The frequency lowering Δ of the normalized cut-off frequency Ω_e in the plated region from Ω_s in the unplated region is calculated as a function of k'_&lt15&gt for the piezoelectric ceramic resonator shown in Fig. 3 (Fig. 6). The Δ remains fairly constant for 0≦k'_&lt15&gt≦k_15. On the other hand, the proportion of elastic effect R_c to Δ can be controlled widely and high R_c may be obtained easily. The frequency spectrum of trapped-energy modes is calculated (Fig. 7). The antiresonance frequency f_a keeps away from the cut-off frequency f_s in the unplated region with increasingc^D_&lt44&gt/ c^&ltD'&gt_&lt44&gt. The design of the difference between f_a and f_s is also investigated for the piezoelectric ceramic resonator shown in Fig. 3 (Fig. 8). In the experiment, the thoroughness of poling was controlled by applying DC electric field between the top and bottom electrodes in the piezoelectric resonator polarized uniformly in the direction parallel to the electrodes. The constant c^&ltD'&gt_&lt44&gt in this process was measured at several different levels of k'_&lt15&gt, in comparison with the calculated values obtained from Eq. 16 (Fig. 9). Response of trapped-energy resonators shown in Fig. 10 was measured at several different levels of k'_&lt15&gt (Fig. 11). the antiresonance frequency keeps away from unwanted response with decreasing k'_&lt15&gt. Furthermore, response of monolithic filter shown in Fig. 12 was measured at the level of k'_&lt15&gt&ltk_&lt15&gt, in comparison with that in case of k'_&lt15&gt=k_&lt15&gt (Fig. 13). The attenuation level is improved in the higher stop band of the filter with decreasing k'_&lt15&gt.

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