The diffracted sound field arround a many-sided barrier or pillar for the incidence of a spherical wave is presented by an application of the general spirit of Keller's geometrical theory of diffraction,which is given by the sum of the singly and multiply diffracted waves produced by the straight edges,as shown in Eqs.(3)〜(6). For the single-edge diffraction by each wedge, Pierce's approximate expression-Eqs.(7)〜(9) based on Macdonal's rigorous solution are used and Pierce's equation for the double-edge diffraction, which is derived by an appication of Keller's theory and his first approximate expressions-Eqs. (7) and (8) for the single edge diffraction, is applied for the multiple edge diffraction as shown in Eqs. (16) and (17).It is shown in Figs. (7)〜(9) that if the distances from each edge to the neighbouring edges, a source and an observating point are larger than a half wave-length, the experimental results are in excellent agreement with the numerical results by the above method where Pierce's second approximate expression Eq. (9) for wedges is used.Consequently, by applying this method, we can predict and control very accurately the propagation of noise arround a thick barrier or a building or a slit or a group of those obstacles etc..

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The effect of the following two factors of fluctuating noise on the annoyance was investigated by means of fluctuating noise (white noises) artificially generated:One is the standard deviation of sound level (σ), the other being the frequency of level fluctuation which is called 'fluctuation frequency' here. Randomly fluctuating noises of Markoff series type were used as principal stimulus sounds to investigate the effect of σ on the annoyance. The level fluctuation of these sounds (Fig. 1) follows the Markoff process and its correlogram is expressed by an exponential function (Eq. 5). The distribution of sound level was normal, and σ took three values (Table1). The number of stimulus sounds including four steady noises was nineteen. The levels of fluctuating noises were set so that either the arithmatic mean of sound level (L^^^-) can be 70±0. 5 dBA or the energy mean level (L_<eq>) can be 87. 5±0. 5dBA. (when σ≒15 dB and L^^^-≒20 dBA, then L_<eq>≒88 dBA. )Three sounds were generated for each combination of σ and L^^^- or L_<eq>. Test was made by the method of magnitude estimation. Students of both sexes, fifteen each, with normal hearing acuity served as subjects. They were given abundant practice before the experiment, and asked to form their judgments on the basis of the total over-all effect of sounds. The data were first normalized by equalizing the geometric mean judgement of each subject, then the numbers were multiplied by a factor to make his or her estimates for the steady noise of 75dBA average one. From the results for steady noises (Eq. 6, Fig. 2), the average estimate of a sound can be converted into the sound level of steady noise which gives an annoyance equivalent to that of the sound. The level is called here Effective Sound Level(ESL). The experimental result shows that ESL increases with σ when L is constant, but slightly decreases with increasing σ when L<eq> is constant(Fig. 3). ESL is in approximate proportion to L<eq> but is neither in proportion to NPL nor to TNI(Figs. 4-6). Similar experiments were carried out to investigate the effect of fluctuation frequency on the annoyance using sinusoidally fluctuating noise (Fig. 7) and randomly fluctuating noise of the Yule series type (Fig. 10). The level fluctuation of randomly fluctuating noise of the Yule series type follows the Yule process (Eq. 10). The power spectrum of this process has a peak at a frequcncy (f_0) and is cut off in the higher frequency region (Fig. 9). Some of the physical characteristics of the stimulus sounds are shown in Tables 2 and 3. The experimantal results (Figs. 8, 11) show that the fluctuation frequency has little effect on the annoyance over the range from 0. 1 to 1Hz. All the results including those of similar experiment using serrately fluctuating noise are plotted against σ for L^^^-=70 dBA (Fig. 12) and for L_<eq>=88 dBA (Fig. 13). It can be seen again that ESL increases with σ when L^^^- is constant, but does not increase therewith when L_<eq> is constant. High correlation coefficients were found between ESL and L_<eq>, L_<10>, and L^^^-+σ, but the correlations between ESL and NPL, TNI, L_<50>, and L^^^- were found poor (Figs. 14-16, Table 4).

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We are well aware of the fact that whole informations on the statistical property regarding the state variables of a stochastic environmental system (such as noise and vibratory nuisace system) can be in principle derived by finding firstly the multivariate joint probability density function of their variables. Moreover, the irregular signals (e. g. , street noise, machine or structure vibration) often appearing in actual engineering fields exhibit various kinds of probability distributions which are not necessarily limited to the usual Gaussian distribution due to the diversified causes of fluctuations. In such cases, a unified expression of the c. d. f. is required, by which the fluctuation mode in course of time of the individual irregular phenomenon under consideration does not influence too much on the whole functional form but is concretely reflected only in its internal parameters. Then, it is required to choose statistical expansion series expression, the expansion coefficients of which reflect the first and higher order statistical concepts which are necessary to express the phenomenon. From the standpoint of the convergence property of expansion expression to be used, the kind of c. d. f. which is chosen for the first term of the expansion expression is of vital importance, since this term describes the principal part of the irregular fluctuation. Furthermore, in practical applications, since a statistical expansion expression is inevitably employed in the form of a finite number of expansion terms, the exact correction to the truncation error is always important. From this point of view, the unified explicit expression of the joint probability density function for the state variables of various kinds of noise and vibratory system is given in the general form of statistical orthogonal or non-orthogonal expansion series with remainder term through the generalization of our previous result on the joint probability density function in a form of multivariate statistical Hermite or Laguerre series expansion and the expansion expression of joint probability density function by P. I. Kuznestov et al. by using multidimensional Hermite polynomials and quasi-moment functions. Hereupon, the joint probability density function which can be arbitrarily chosen in advance for the convenience of noise and vibratory nuisance analysis is taken into the first term, and the effect of the fluctuation pattern of state variables on the joint distribution form is concretely reflected in the second and higher terms in the above expansion expression. In a concrete form, when irregular noise or vibration Z(t) of arbitrary distribution type can be considered as the sum of two different irregular processes X(t) and U(t) as a result of the natural internal structure of the fluctuation or the analytically artificial division of the fluctuation, unified statisticalt treatment for the c. d. f. of the resultant irregular fluctuation Z(t) (=X(t)+U(t)) is introduced exactly in the form of finite expansion terms (here, X(t) and U(t) may be mutually correlated and need not always show Gaussian type distribution). First, let us introduce an arbitrary function ψ(Z)(ZΔ___=(Z(t_1), Z(t_2), …Z(t_k)) with the property of Eq. (1) and consider its expectation value, Eq. (2). Applying the multi-variate Taylor series expansion with a remainder term to Eq. (2), Eq. (3) can be obtained. Our main problem is how to derive the multi-variate probability density function P(Z) in the form of finite expansion terms based on statistical informations of X(t) and U(t). After somewhat complicated derivation, we obtain the expansion expression, Eq. (12). Next, multi-variate joint moment <Z^<m_2>_1 Z^<m_2>_2…Z^<m_k>_k> is derived in the general form (cf. Eq. (20)), which can be connected with power, bi- and poly-frequency spectra. Furthermore, we have experimentally confirmed the validity of our theory

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