We presented previously a report on processing of aircraft noise around airports with an electronic computer. In this paper, we report about the results obtained by analyzing factors considered closely associated with aircraft noise using the method of factor analysis based on the quantification theory. The method of factor analysis permits assignment of the most suitable values to the categories of factors such as type of aircraft, take-off or landing which can not be represented by numerical values unless some special measures are taken. This process, therefore, makes it possible to accomplish various multidimensional analyses and to predict such values as noise peak from factors such as type of aircraft, take-off or landing and flight altitude. Factors analyzed are as follows:(1)Noise peak value[dB(A)]:Noise peak value in excess of 70dB(A) (2)Noise duration(sec):Duration of noise above 70dB(A) (3)Type of aircraft (4)Take-off and landing (5)Flight altitude(m) (6)Wind direction was also examined in addition to the above factors. The observation points are A and B shown in Figure 1. Regarding the wind direction, in addition to the above, C and D are used as observation points. Table 1 shows a part of the sample data. The noise peak value and noise duration are taken as external data Table 2 shows an example of results obtained by analyzing the various factors mentioned previously using the method of factor analysis. X is a value assigned to each factor category corresponding to type of aircraft, take-off or landing and altitude. The above-mentioned external data can be predicted from X, which can be obtained from the following equation:FX=A^*(1) In equation (1) the matrices F and A^* are obtained from the sample data shown in Table 1. In order to predict the external data from the results shown in Table 2, it suffices to add X, the value assigned to each factor category, to the mean value X^^^-. It should be noted here that X has been normalized in such a manner that the mean value of each factor becomes zero. For example, in case three categories;F4 Phantom, Take-off and Altitude of 601m or more are given(Sample No. 1 in Table 1), the noise peak value can be predicted from Table 2 as follows:6. 16+2. 39+(-0. 92)+83. 36=90. 99 dB(A) In this particular case, the actual observed value was 91 dB(A). The noise duration can be predicted in the same manner. The accuracy of analysis, namely, accuracy of prediction is represented in terms of a multiple correlation coefficient. Figure 4 shows the value calculated from the results of observations made at point A and indicates how the multiple correlation coefficient varies as 3 kinds of factors are added to external data one by one. The same tendency is also noted at point B, In Figure 5, the partial correlation coefficients of individual factors corresponding to the noise peak values at both point A and point B are shown. Weights of various factors at point A and point B related to the external data are compared with each other in Figure 6 and Figure 7. These weights correspond to the ranges of the factors such as those shown in Table 2. The weight and the partial correlation coefficient, as a rule, have the same tendency. Use of weight, however, is more convenient than the presentation in the partial correlation coefficients, for it permits direct comparison between physical values. Figure 9 shows how the external data are affected by the wind direction. It is known that the head wind and cross wind affect the external data positively, while the tail wind affects them negatively. Since the preliminary investigation has indicated that the effects of temperature and humidity on the noise peak value may not be negligible, this point deserves further investigation. The calculation of all the above statistics was made by using Hitac 8500.
A holographic side looking sonar is thought to be used for mapping sea-bottom from the surface. The principle of the holographic side looking sonar is the same as that of a side looking radar with optical data processing. For conventional mapping methods a two dimensional map can be obtained by either a complicated scanning of a transducer with sharp directivity or by a simple scanning of a transducer with broad directivity upon reducing angular resolving power. For the holographic side looking sonar, however, only a simple scanning is necessary because the resolving power increases when the directivity becomes broad. This method, therefore, seems to be useful for practical operations of sea-bottom mapping. A holographic side looking sonar is experimentally studied by using a transducer with resonant frequency of 455kHz in a water bath(shown in Fig. 4)and reconstructed images are obtained by a He-Ne laser. (Reconstructed images and their originals are shown in Figure 6)The experimental result shows the possibility of the practical use of the holographic side looking sonar for mapping sea-bottom from the surface.
In the measurement of a piezoelectric vibrator having a small figure of merit by the resonance-antiresonance method, the high minimum-to-maximum-admittance-ratio is obstructive to an accurate determination of the electromechanical constants. In the present paper, various approximate formulae of correction for the ratio are derived and examined. The present treatment is always based on the through-the-center approximation, as described by Martin. The constants to be determined are K, γ andω_0 in eq. (2). When the values of M and Q(or K andγ)are given, the squares of the reduced frequencies z_m and z_n are determined by solving eq. (9)and the value of r is exactly evaluated. In paragraph 2. 1, z_<m, n> in eq. (11)are used, whereγ^2/K is ignored in comparison with γ^2/K^2 in(10)given by Martin, and thus the approximations (I. 1)〜(I. 4)are obtained. From the graphical treatment on the basis of Fig. 1 are derived the formulae (II. 1)〜(II. 4). The derivation is understood by following the sequence from(13)to(22)and seems most faithful to the through-the-center approximation. Ignoring the term δ_m in(II. 1)and(II. 4)provides the approximation III. Using the frequencies(25)instead of (11), the other sets of formulae can be derived, but are not described here. The accuracies of the present approximate formulae are examined in §3, compared with those given by Martin. In 3. 2, Δ_<mn> and r are calculated using the approximate formulae for M and Q given, and they are compared with the exact solution on the Δ-r chart after Martin, as shown in Fig. 2. Such indications, however, are not suited for examining the accuracy of the approximation. The fractional errors are estimated by(29)for each approximation and the distributions are shown on the Δ-r diagram, as in Fig. 3. The approximation III seems to be the best one. Actually, the squared frequency difference δ_<mn> is not an observable. Substituting (I. 3) for δ_m in(6), we have α_1 in (30), which is measurable and can be used instead of δ_<mn> in the case I. Similarly α_2 in(33)is obtained in the cases II and III. Uually, we often use β or 2Δ_<mn> in place of δ_<mn>. The errors for IIIα, IIIβ and IIIΔ are shown in Fig. 4. When no corrections are taken into account in respect of r, as described in (0β. 1)or(0Δ. 1), the errors for K are not ignorable, as shown in Fig. 6, whereas that of K_<IIIα> is negligible. The errors of the present approximations always increase in the upper left range of Δ-r chart. These defects are not dissolved within the through-the-center approximation. In §4, another approximation is considered rather in an empirical way. Examining the plot of the fractional errors of IIIα versus Δ_<mn> in Fig. 5, new parameters ξ and ζ are introduced for the readjustment errors, as in eqs. (34)〜(37). The errors of IIIα are plotted against ξ or ζ, and these plots suggest a way of decreasing the errors. Thus the improved formulae(*. 1)〜(*. 3)are found. The errors are within 3, 3 and 1% for K^*, γ^* and δ_m^* respectively. Shibayama early reported a graphical method of evaluating small coupling factor. The present approximation covers the entire range of his chart with sufficient accuracy. The proposition of the two sets of the approximate formulae, IIIα and *, is the purpose of the present paper.