The purpose of the present paper is to clarify the effect of sound duration on annoyance, i. e. how the duration effect (annoyance increase per doubling of duration) can be quantified, and whether or not the duration effect depends on the sound level (level-dependent) and/or the length of duration (duration-dependent). Stimulus sound is white noise whose duration ranged from 30 ms to 90 s for each of four levels. Idealized time patterns of signal employed in the judgment test are shown in Fig. 3, and some physical characteristics of the stimulus sounds are presented in Table 1. In order to avoid too long time for one test, all stimulus sounds were divided into three groups, I, II and III, as shown in the table. The data obtained in these experiments can be compared with each other through common stimuli (marked with asterisk). Ten male and ten female students with normal hearing acuity served as subjects. They were given abundant practice before the actual experiment began. The subjects were instructed to judge the whole perceived magnitude of the sound, for example, annoyance, unpleasantness, etc. The sounds were arranged at random with respect to test variables and presented to each subject through a headphone (diotic) in the sound proof room. The subjects assigned numbers proportional to the magnitudes of his perception for the sounds presented one after another. The number denoting annoyance of a sound, the level and the duration of which were 80 dB and 10 s respectively, was arbitrarily equated to one and the other annoyance estimates for each subject were transformed with reference to this value. According to the results shown in Fig. 5, the annoyance increases linearly with the logarithm of the length of duration. The following equation was obtained by multiple regression analysis with two independent variables: logψ=0. 229 logt +0. 0203L-1. 842, where ψ is the ratio of annoyance, t is the length of duration (s), and L is the peak level (dB). From the equation, it can be seen that annoyance becomes twice when the level of sound increases by 14. 8 dB. Using this relation, ψ is expressed as the relative sound pressure level (RSPL), which is shown in the left ordinate of Fig. 5. Taking the interaction between L and t into account, we obtain the following equation: logψ=0. 0018L logt +0. 091 logt +0. 020L-1. 817. Since all regression coefficients in this equation are significant (significant level; 0. 05), the duration effect is considered to be level-dependent. By expressing L as a function of t and ψ in the equation, we obtain equal annoyance contours for some values of ψ (Fig. 10), which show hyperbolic curves because of the interaction term. Therefore, the apparent duration-dependency of duration effect observed in Fig. 10 should be attributed to its level-dependency. Experimental results are also plotted (Fig. 11) against acoustic energy level of the stimulus sound (L_<en> in Eq. (16)). It can be seen that the annoyance increases approximately linearly with L_<en>.
For the analysis and design of acoustical, mechanical and electromechanical systems, the equivalent electrical circuit analogy is widely utilized, which is in particular by favoured by electromechanical engineers. The equivalent electrical circuit results from a process to represent the general three-dimensional system approximately with a corresponding nondimensional discrete network. A Helmholtz' resonator can equivalently be represented by the parallel circuit of a lumped inductor and a capacitor, and an acoustic tube by the distributed transmission line of parallel wires. The former is available for the resonator whose dimensions are much smaller than the sound wave-length under consideration, and the latter for the tube whose diameter is much smaller than the sound wavelength as well. In general acoustic field systems, however, the wave equation is to be solved under proper boundary conditions. The finite difference method is useful for the numerical analysis. Arai developed the equivalent electrical circuits based on the finite difference method and investigated acoustic filters and sound-absorbing wedges by use of the corresponding electric simulators. Now the finite element method is being high-lighted. In this paper are presented equivalent electrical circuits for acoustic fields based on the finite element method. A fundamental equivalent electrical circuits is developed for a liner triangular or tetrahedral acoustic element (Fig. 4 and 5), from which the equivalent circuit for an arbitrary sound field can be formed simply by connection. As is always the case, the treatment at the boundaries is simpler than that of the finite difference method. When the equivalent circuit is once developed, its analysis is usual by possible not only in the frequency-domain but also in the time-domain as an impulse response.
In this paper, a calculation method for the amplitude of reflected sound waves from a spherical body at a short distance from the sound source is described. This method is expressed in the terms of the "directional sound reflectivity of the spherical body". This directional sound reflectivity is the ratio of the sound pressure of the reflected wave from a spherical body at the position of the receiving point to the particular pressure which would be calculated by the theory of geometrical optics. As shown in Fig. 1, a case is considered in which the spherical body is put on the origin of spherical coordinate. V_N, that is the component of the particle velocity of the incident wave in the radial direction at the surface of the body, is given by Eqs. (4) and (5). Reflected waves can be expressed by Eq. (6), which is the solution of Laplace's equation. The particle velocity of the reflected wave at the surface V_R must be -V_N. So, the coefficients C_n are given by Eq. (8). The velocity potential of the reflected wave from a spherical body can be written by Eqs. (9) and (10). If the sound wave can be treated by the theory of geometrical optics, the approximate value of the value potential of the reflected wave is expressed by Eq. (15). From the above relations, the absolute value of the directional sound reflectivity of a spherical body, R_<ps>, is given by Eqs. (16) and (17). This relation becomes Stenzel's relation when L=M and M is increased up to infinity. Figs. 3, 4 and 5 show examples of the numerical values of |R_<ps>| when L=M. Some experiments were performed in the air at a frequency of 40. 00 kHz so as to verify the validity of above theories. Fig. 6 shows the arrangement of the apparatus in the experiment. Two piezoelectric transducers were put toward a steel ball. The pulse width of the sound wave from the transmitter was 0. 4 ms. The amplitude of the reflected pulse was measured on the screen of the C. R. O. Fig. 7 and 8 show examples of these experiments. In these experiments, the results qualitatively agree with the theory when the angle between the direction of incidence and that of reflection is comparatively small. From the results of above experiments, it is found that the calculation method for the reflection of sound waves from a spherical body and its numerical results are useful when the sound source is at a short distance from the spherical body.