Some acoustical characteristics of the spherical cavity of the sound source used as a standard for free field measurements have been investigated. This sound source consists of a spherical shell and of a diaphragm vibrating at a constant rate at the circular window on the surface of the spherical shell. In general the medium in the cavity makes the back load of the diaphragm, and its impedance which affects the vibration of the diaphragm can not be neglected;so the necessary conditions for designing the standard sound source should be investigated. A number of papers on the spherical sound source have been reported;most of them, however, have been described on the radiated sound field, and only a few have been concerned with the sound field in the spherical cavity. To design a standard sound source of greater stability and reliability, the back load impedance of the diaphragm which is caused by the sound field in the cavity should be made clearer. In this paper the acoustic load of the diaphragm is analyzed and experimentally verified, and the sound pressure distribution in the cavity is also calculated. It has been shown that the frequency characteristics of acoustic impedance which changes over a wide range are not desirable for the formation of a standard sound source. To avoid these undesirables sound absorbing materials have been inserted into the cavity and the acoustic impedance resulting from two different packing methods have been analyzed. As a result, it has been shown that the undesirable phenomena can be eliminated and that the values of acoustic impedance can be limited within a narrow range.
The characteristic frequencies of, a special bell with 34 horns on the upper half(Fig. 1)originated by artist T. Okamoto, and those of an ordinary Japanese bell attached with a brass bar(Fig. 4)are studied. The spectrum of the characteristic frequencies of the special bell(Fig. 2)and the continuous changes of the frequencies of it by the change in the weight on one of the horns are measured(Fig. 3). The frequency spectrum has about 50 lines under 1 kc/s, which are about 10 times as many as that of the bare bell. The frequency spectrum of the special bell is in accordance with the continuous changes of the characteristic frequencies. Since, by the increase in the weight of the horn, the values of the characteristic frequencies generally decrease, those of the bell with horns will decrease below those of the bare bell with the addition of horns. As a simplified model of the special bell, a brass bar (circular in cross section) is attached to an ordinary Japanese bell(Fig. 4), and the continuous changes of characteristic frequencies by the change in the length of the bar are measured(Fig. 4). The steep descending part of the curves is represented by formula (1), approximately. Since the formula for the fundamental frequency of the lateral vibration of the attached brass bar as a canti-lever is represented by (2), the steep descending part of the curve is intimately related to the lateral vibration of the attached bar. On the other hand, the lines parallel to the l axis represent closely the characteristic frequencies of the bare bell. The experimental curves pass close to the above mentioned curve (1) and the parallel lines, which cross each other. However, the experimental curves do not cross each other and these properties are explained by the theorem derived by Neumann and Wingner qualitatively. By the above analysis of the experimental curves of Fig. 4, we arrive at the conclusion that the continuous changes of characteristic frequencies of the special bell with horns and those of the Japanese bell attached with a bar may be interpreted as a coupling phenomenon between the bare bell and the attached horns or bar. As a model of the special bell with horns which allows mathematical analysis we shall next consider the vibrations of the canti-lever attached with a thin bar.
As a model of the special bell with horns and of the Japanese bell attached with a bar, which permits mathematical analysis, a canti-lever attached with a thin bar is considered(Fig. 1). The kinetic and potential energies of the system are represented by the coordinate system shown in Fig. 2 (Eqs. 3, 3', 4, 4'). Expanding the deformation y_I, y_<II> of the system by the eigen-functions of canti-levers (Eq. 9), the kinetic and potential energies are represented by the coefficients q_<Ir>, q_<IIr> of the expansions. (Eqs. 12. 12', 13, 13'). Thus, Lagrange's Eqs. of motion are established and Eqs. (17, 18) which determine the vibration of the system, are obtained. These are linear differential Eqs. with constant coef. of ∞ degree of freedom. The characteristic frequencies of the system could be obtained in principle, by the ordinary method. The procedures are shown briefly from Eqs(19) to(35). The difficulty arises from a practical point of view due to the extent of the degree of freedom. However considered physically, the higher mode of the expansion may be neglected, but to solve the Eq. (33) rigorously is not easy. An effort is made to get out of the difficulty by the use of the perturbation theory. But, since these results do not agree with the experimental results, preference is given to solve the Eq. (33) by the computer (IBM 1130) using Jacobi's method. The order of symmetric matrix, the elements of which are defined by Eqs. (32, 32', 32"), is limited to 10. The results of the calculations are shown in Table 4 and Fig. 6. The agreement with experimental results is valid up to 1%. Because the above calculations are too formal, it is next shown that the nature of the Eqs. (17, 18) has the characteristic feature of the continuous changes of frequencies of the special bell and the Jap. bell attached with a bar, shown in the Fig. 3, 4, of the previous article. Firstly, if the terms containing μ'_<II> in the Eq. (17) is neglected, and μ'_<II> L_<II> in Eq. (18) is deleted, Eqs. (37, 37') are obtained. The Eq. (37) shows that q_<Ir> vibrates with angular frequency ω_<Ir> and by inserting the results of Eq. (37) in (37'), the Eq. representing the typical forced vibration is obtained. The Eqs. shows that the system has the characteristic frequencies ω_<Ir> and ω_<IIs> (L_<II>) approximately (Fig. 3), which are the characteristic frequencies of the bare canti-lever and those of the thin bar as a canti-lever. The curves which represent the characteristic frequencies of the system as a function of the length of the attached bar should pass close to these frequency curves. These characteristics of the Eqs. (17, 18) may be foreseen as those of the special bell and the Jap. bell attached with a bar. Next, the perturbation treatment is mentioned briefly in Eq. (33), which shows the behavior of the continuous changes of the frequencies where the curves approach each other (Fig. 4, 5). Finally, the calculated results by the computer and the experimental results are shown. At the end of the article, the beat phenomena observed at the part where the curves approach each other are touched on and their relation to the past studies of the author is explained.
Even in the steady part of speech or musical tone, small fluctuation is superposed on regular repeated waveforms, and it has been clarified that these irregularities play an important role in the perception of sound. In general, the sound which we usually hear, includes some random fluctuation. These are still unclear points, however, about the perception of such sound. As a step to interpret these unclear points, we determined the discrimination limit of pitch associated with pulse trains as a function of the pulse frequency and of temporal and amplitude jitters. The results obtained are as follows : 1) Jitter in amplitude has little effect on pitch discrimination, while jitter in pulse intervals has large influence on pitch; for the lower frequency the effect of the jitter on pitch discrimination is greater than that for the higher frequency, and even a small amount of jitter makes the discrimination worse in a great deal (Fig. 3). 2) Random fluctuation in amplitude has large effect on detectability of temporal jitter (see Fig. 6). 3) The above results, 1) and 2), can be explained by the power spectrum of a pulse train (see Fig. 4 and Fig. 5). 4) In the case where the band-limited pulse trains including only the fundamental component were used as stimuli, there is little difference in pitch discrimination from the case without band limitation. This result shows that the former contains sufficient information on pitch (see Fig. 7). 5) When the band-limited pulse trains including only the second harmonic were used, pitch discrimination rapidly deteriorates with the increase of jitter in pulse interval. This result is due to the fact that the power of the second harmonic of the pulse trains including jitter, rapidly decreases with an increase of jitter (see Fig. 7). 6) When pulse trains with band limitation within 2 octaves including the 4th and higher harmonics were used, fairly good discrimination of pitch was obtained in low frequency (below several hundred Hz), but in higher frequency the pitch discrimination is scarcely possible. The upper frequency limit capable of pitch discrimination is about 800 Hz, and this frequency is considered to correspond to the upper limit of periodicity pitch perception. This value is regarded as valid in consideration of the psychophysical and physiological evidences reported by many authors (see Fig. 7 and Fig. 8).