As shown in the previous papers, the beat frequency, that is, the difference between the high and low frequencies of the fundamental tone, was freely adjusted by means of loading off the surface of the Japanese bell, the clearness of the beat has not yet been discussed. Clearness is achieved by placing the striking point in the middle of the high and low nodal lines of the fundamental tone. Therefore, an experimental research is presented in order to investigate how the nodal lines of the Japanese bell are formed. A Japanese bell, so-called "Hansho", was specially made with forty-eight nipples, so called "Nyuto", on its surface. The bell was set in resonance by means of a vibrator controlled by an oscillator, and the natural frequency of the bell was measured by a frequency counter and the nodal lines were determined by comparing the phase difference of two signals detected by two pick-ups. The state of vibration at each point on its surface was recorded by a level recorder through the pick-up. First, the high and low frequencies of the fundamental tone were forced to be unified by means of paring off the nipples. Then, the two nodal lines became to be one and the beat did not occur. Next, when the load was applied to this none beat bell, an experimental measurement was calculated to find how to be formed two nodal lines formed the fundamental tone. Moreover, the relationship of the clearness of the beat, the striking point and the nodal lines was investigated. The results are as follows. (1) The bell can be adjusted so as to have only one frequency in each partial tone. Then, one nodal line of the bell comes into existence so that the striking point always becomes the loop. (2) When the load is applied on one point, two nodal lines generate, one of which corresponds to the high frequency and the other to the low one. At this time the loaded point becomes the node of the high frequency and the loop of the low one. (3) When the load is applied on two points, the nodal line of the high frequency is located in the middle of the two loaded points and the low loop line in the middle of that. (4) When the striking point is located in the middle of two nodal lines, the contrast of the beat becomes the greatest. Therefore, the clearness of the beat is maximized. As a result, the striking point ought to be located in the middle of two nodal lines of the fundamental tone in order to make a bell with the greatest beat contrast.
In the previous paper, starting from a criticism of the Rice representations of normal random noise and generalizing a random noise model in terms of a trigonometric series consisting of uniformly almost periodic (U. A. P. ) functions, a new trial for a mathematical formation of a random noise model was theoretically introduced by using U. A. P. functions consisting of arbitrary component waves. That is, the new mathematical model I_N(t) (cf. Eq. (1)) is characterized by(1) No necessity for introducing any probability distribution law for either amplitude or phase at the outset, and(2) Gaussian distribution characteristic formed automatically in the course of time. Keeping in mind the variety of the component wave form, the arbitrariness of the number of component waves and the complexity of mathematical expressions involved and their statistical treatment, the use of the digital simulation technique is inevitable for the experimental confirmation and is the most successful way for the present study. In this paper, it is experimentally pointed out on the normal probability papers (cf. Figs. 7〜10 and Figs. 13, 14) from various points of view that a cumulative probability distribution Q(I_N) (cf. Eq. (3)) is asymptotically a standard normal distribution as N tends to infinity and the choice of the component wave F(θ) contributes substantially to the speed of convergence tending to Gaussian distribution on several concrete cases where the component wave has respectively the specialized forms (cf. Cases I〜V in §2. 1 and Figs. 7〜14). Furthermore, the characteristic that the nth order moments (n=4, 6, 8) of I_N(t) become asymptotically those of standard Gaussian distribution for a sufficiently large value of N is experimentally confirmed in a special model of random noise formed in terms of trigonometric series consisting of U. A. P. functions(cf. Fig. 15).