Sound radiation from a sound source of concave shape is analyzed by applying wave theory. In our analysis, we divide a sound source into thin cylindrical subsections whose radii vary gradually, so that the propagation of a plane and higher modes of sound waves in each subsection is calculated in a stepwise manner by solving boundary value problems. It may be expected that the solution of this analysis is applicable to prediction of the sound field radiated from a cone type or a horn type loudspeaker in an infinite baffle. In this paper, some numerical calculations are made for the sound fields of a rigid piston located at a certain depth from an infinite baffle (Fig. 5)and of a rigid cone in an infinite baffle (Fig. 8), and are compared with the values obtained by the usual approximate methods. Fig. 6 shows the dependence of the normalized sound pressure, which is defined by Eq. 50, upon the frequency parameter ka_1 for the case of the rigid piston. In this case, comparing the values obtained by a plane-wave approximation, we may be able to say that they are in agreement with each other in some degree. For the case of the rigid cone, however, there are significant differences between the values by a geometrical acoustic approximation as shown in Fig. 9. The values by the present method fluctuate around the values by the approximate method.

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The loudspeaker with flat diaphragm as vibrating plate may have many superior qualities, but there occur resonances in the comparatively low frequency range because of lack of rigidity of flat diaphragm. Advantageous for suppressing the resonances is a 'multi-drive' loudspeaker, of which diaphragm is driven by plural voice-coils, and the driving forces are adjusted so as to cancel the resonances. If the diaphragm is driven by a voice-coil(or voice-coils) attached to the nodes of an undesirable vibration mode, that vibration mode will be small because of a large driving-point impedance at the nodes. However, it is necessary to find the effect of the mass of voice-coil on the eigenfrequencies and eigenfunctions of other vibration modes, for example, in order to determine the ratio of driving forces. Eigenfunctions of a circular plate with a voice-coil(or voice-coils) are expanded by the well-known eigenfunction {P_m(r)} of a plate with no voice-coil. After several operations, eigenfrequencies{ω_&ltcm&gt} of the vibration of a plate with a voice-coil are expressed in the form of matrix, as a function of the eigenfrequencies {ω_m}, values of eigenfunctions at voice-coil positions P_m(r_1), and the mass of voice-coil to mass of plate ratio, of a circular plate with no voice-coil. Eigenfrequencies of a circular plate with plural voice-coils can be computed with considerable ease, and it can be seen from the matrix expression that, when the voice-coils are attached to nodal circles of the vibration modes of a circular plate with no coil, the eigenfunctions and eigenfrequencies of that mode do not change at all. We calculated eigenvalues k_ma of a circular plate with a single voice-coil of various radiuses and masses, then calculated the eigenfunctions of a circular plate with one or two of voice-coils attached to the nodal circles of the plate. In addition, we calculated eigenvalues of an annular plate with a aingle voice-coil of various radiuses and masses, then calculated the eigenfunctions of an annular plate with a single voice-coil, attached to nodal circle thereof. The results obtained show that, when the position of the voice-coil goes far from the nodal ciecle, eigenvalues decrease, proportionally to the mass of the voice-coil and the displacement of the plate at the voice-coil position. In a circular plate and an annular, plate, free at their boundaries, the law of `Conservation of Momentum' is satisfied, including the momentum of voice-coil. That is to say, eigenfunctions make such a change that the change of the momentum of the plate cancels that of momentum due to the mass of voice-coil. Thus, the larger are the mass of voice-coil and the displacement of the plate, at the voice-coil position, the greater is the change of eigenfunctions. The force-ratio of the voil-coils to suppress the first and second resonances simultaneously, is obtained from the displacements of the first resonance mode at the position of the secondary nodes of the plate. The mass ratio between the two voice-coils can be selected in such a manner that the first eigenfunction becomes nearly equal to that of the plate with no voice-coil.

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Several types of non-linear distortions of audio nstruments are discussed sytematically using calculation models. The model of an input pulse shown in Fig. 1 is characterized by an asymmetric wave form with zero DC component. The pulse responses corresponding to non-linearities, viz. , the S-type non-linearity(Fig. 2), clipping(Fig. 4) and cross-over distortion(Fig. 6) of an amplifier, the transient distortion of a level-compressor, and the S-type non-linearity(Fig. 9) of a loud-speaker, are illustrated if Figs. 3, 5, 7, 8, and 10 with their spectra, respectively. Since the growth of the DC component is typical of the non-linearity of an amplifier, which is in constrast with the case of a loud-speaker, it can be said that the non-linearities of an amplifier and a loud-speaker are physically distinguishable. It thus appears that one can judge an amplifier by hearing a musical sound through a loud-speaker.

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In this paper, a method of harmonizing both the second partial F_2 and the third partial F_3 of the cantilever vibration with the fundamental F_1 is proposed, in order to improve the musical sound of a cantilever-electro-piano. The cantilever is loaded with a small mass m at Kl(fig. 4). The equations of the motion of the cantilever with a mass are set up as equations(1)〜(9), and the solutions are shown in Fig. 5. The conditions in which F_2 and F_3 are equal to integer multiples of F_1 are indicated in Fig. 7 and Table 1, the results of experiment in Table 2, and the figures of various kinds of the cantilever-electro-piano will considerably be improved with these new cantilevers.

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This paper deals with a simple method for measuring accurately the equivalent circuit elements of a piezo-resonator. This new method is especially very valid for such a piezo-resonator with low resonant quality factor Q and small figure of merit M that the free input admittance curve shows a distorted small circle or does practically no circle in the conductance G and susceptance B plane. It is understood that the method can be applied to measuring equivalent circuit elements of a piezo-resonator with high Q or medium Q. It is found clear from measured results that the equivalent circuit elements of the resonator with low Q and M can be determined by the new method with same order of exactness as for a high Q or medium Q resonator.

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In this paper a new design method of a phase inverter loudspeaker system, which uses for the first time the non-linear optimization method, is discussed. From circuit analysis of Fig. 1 (b), the sound pressure transfer function of a phase inverter loudspeaker system can be obtained as shown in Eq. (1) and its frequency response function G(X, ω) is given in Eq. (3). The frequency response of G(X, ω) is controlled by the system parameters vector X(Eq. (2)), whose elements X_i, i=1, 2, ・・・, 8, are composed of r_0, s_0, m_0, r_B, s_B, r_P, s_P and m_P, respectively, in Fig. 1(b). In order to realize flat response and at the same time extension of the low frequency response, we define an evaluation f(Eq. (4)). The value of f decrease as G(X, αω_i) approaches to the target-level G_i, where α is a positive parameter and has a function of extending the low frequency response, and where ω_i is the freqency point of target. By minimization (optimization) of f we can obtain X which realizes flat response and extension of the low frequency response. In Fig. 2 one point of evaluation is considered to explain the process of extending the low frequency response. Point A on the initial response reaches point B whose level is close to the target-level G_1 after k_1-th trial of minimization. Then the value of α is changed from the initial value α_0 into α_1(α_1&ltα_0), and the evaluation point moves to point C from point B. Again point C reaches point D whose level is close to G_1 after k_2-th trial of minimization. The value of α is again changed from α_1 into α_2(α_2&ltα_1), and the evaluation point moves to point E from point D. The value of element X_i has the upper and lower limits within which X_i varies, or a specified value. This constraint on the value of X_i is decided by a designer's intension, and makes the process of minimization complicated. Therefore X_i is transformed into Y_i which has no constraint as shown in Eq. (6) and Fig. 3, and f is minimized, by the Davidon-Fletcher-Powell method, with respect to Y whose element is Y_i. An example of the minimization process is shown in Fig. 4 for two variables. When Y realizing minimization of f is obtained, Y is transformed back into X in Eq. (6). The flow diagram of this design method is shown in Fig. 5. A model of evaluation function vs. the number of trial times of minimization is shown Fig. 6. In this paper three design examples are given. The constraint and the frequency response of target for the design examples are given in Eqs. (19) and (20), respectively. Design example 1 is for a drone cone type system in the case when all elements of X are variables (Figs. 7, 8 and Table 1). Design example 2 is for the drone cone type system in the case when the volume of box is specified (Fig. 9, Table 2). Design example 3 is for a port type system in the case when the driver speaker is specified (Fig. 10, Table 3). This design method makes it possible to realize flat response and at the same time extend the low frequency response under the constraints.

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This paper concerned with the vibration analysis of a cone loudspeaker by the finite element method (FEM). The factors which influence the frequency response of the sound pressure level (SPL) of a loudspeaker are disccused in detail. We used NASTRAN as a program for FEM. Our automatic data generator is employed in order to simplify the engineers' task of input data preparation. The frequency response of SPL is obtained by integrating the velocity of each node. A storage graphic display is used to show the shapes of the characteristics of loudspeakers. The calculation of SPL requires about 18 minutes at CPU time with an IBM 370/148 computer. We consider the influences of the cone, the surround (outer suspension), the spider(inner suspension) and the voice-coil bobbin on SPL of a cone loudspeaker. In this caalculation the parts are divided into axisymmetric conical shell elements. The calculation error of the maximum peak frequency of SPL (f_h) is 6% in Fig. 4. First, we considered the influences of the shapes of cones. The first-order eigenvalue (f_1) of a 0. 17mm thick conical cone with a semi-apex angle of 68 degrees is three times that of a 0. 17mm thick curved cone with a curviture of 0. 046m. The thickness dependence of f_1 is shown in Fig. 6. f of the curved cone varies with the thickness of the cone. However, f_1 of a conical cone with a small semi-apex angle is hardly affected by the thickness of the cone. This implies that f_1 of the courved cone depends on bending vibration. Second, we discussed the influences of the surround. The peaks and dips of SPL from the surround are caused by natural resonances. The maximum peak frequency of the tested cone suspension is about 5 kHz. As is shown in Fig. 10, this resonant frequency is inversely proportional to the radius of the curviture of the surround and proportional to the sound velocity in its material. Third, the influences of the voice-coil bobbin and the inner edge of a cone are discussed. The thickness, elasticity and length of the voice-coil bobbin affects the frequency range of SPL. Finaly, the resonance of the spider is shown in Fig. 13. The first-order resonant frequency of the tested spider is shown in Fig. 13, and about 777Hz. The measured vibration pattern fairly agrees with the calculated one. This resonance of the spider decreases the smoothness of the frequency response of SPL.

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