The effect of a plane discontinuity on a plane wave propagating in a cylindrical tube of arbitrary cross section is, in some sense, analogous to the role of a transformer in elctrical circuits. It has been widely used in designs of acoustic devices as a transformer element. In this paper, detailed investigations based on wave theory have been made on an acoustic transformer with discontinuous reduction in circular cross section with rigid wall, as shown in Fig. 1. This is because it has a wide range of applicability as well as a merit of rather simplified analysis. The usual treatment of this sort of problem in acoustics assumes only a simple plane wave propagation in the tube, and it was J. W. Miles who made the first rigorous formulation by considering higher order modes excited at the discontinuity. Thereafter he and F. C. Karal contributed to the equivalent circuit representation of an acoustic transfomer. Their investigations, however, were chiefly concerned with formal and qualtative analysis. In this paper, therefore, rigorous and quantitative analysis has been made and several examples of the pressure amplitude distributions along the tube axis are shown in Fig. 4, where the plane wave, exp i(ωt-kz), is incident to this element with diameter ratio, ξ=1/√<2>. By examining this figure, we may obtain the important information about the effects of the higher order modes in the vicinity of the discontinuity. Fig. 5 shows the manner how the incident power is divided between transmitted and reflected powers. Also, systematic researches on simple plane-wave approximation and plane-piston approximation have been made in order to clarify their physical meanings and characteristics as well as their mutual relations. In Figs. 6 and 7 approximate solutions are compared with the accurate ones with respect to input impedance, transmission and reflection coefficients. In these figures, the results of plane-wave approximation correspond to the value of ka=0. And plane-piston approximation gives rather good results as far as (0, 0) mode is concerned. By referring to these figures, useful information on the design of this element and on the limitation of each of the approximate methods may be obtained.
In constructing a mathematical model of random noise, what is indispensable is first to define a suitable deterministic expression to describe a time-sequence of random phenomena according to the law of causality and then to incorporate into the mathematical framework the concept of probability distribution which accounts for the stochastic character existing in actual random phenomena. More explicitly, the latter concept of probability expresses the possibility that the random process may assume any value in a specified interval at an arbitrary time. An important problem is how to unify the deterministic character and the probabilistic character of the random noise. We must notice the fact that in random processes existing in the physical world all the possible varieties of amplitude, phase, or other physical quantities really appear in a sufficiently long interval of time. However, the Fourier-series representations (1) and (2) of white noise due to S. O. Rice do not seem to give an organic unity of the deterministic and stochastic characters of random noise, since the probalility distributions are assumed at the outset independently of the lapse of time. In Eqs. (1) and (2) the frequency ratio of fundamental frequency to another frequency is always a reciprocal of an integer. This seems to be the very reason why the probability distribution had to be assumed independently of the lapse of time. In this paper, it is shown that a new mathematical model of random noise can be formed in terms of a trigonometric series consisting of uniformly almost periodic functions as follows I_N(t)= ��^^^N___<n=1> c_n cos2π(f_nt+φ_n) Where all the frequency ratios such as f_1/f_2, f_2/f_3, form a set of irrational numbers. Here C_n=C_0 for all n but small degree of inequality among C_n is allowable. Now, it is not necessary to introduce any probability distribution law at the outset into the new model, because probability distribution is automatically formed in the course of time according to the quasi-ergodic hypothesis of P. & T. Ehrenfest. If a white noise is regarded as the sum of outputs of infinitely many independent oscillators having different frequencies as was pointed out by A. Einstein and L. Hopf, it may be more natural to think that all the frequency ratios are irrational numbers than to think that they are integers, because the set of all irrational numbers are exceedingly dense compared with that of all integers so that the probability that a frequency ratio happens to be an integer is far less that the probability that it is an irrational number. Hence, the new model of white noise seems to explatin the actual structure of random noise better than the Rice representations. Furthermore, in the light of the method of steepest descent due to P. Debye (cf. Eq. (12)) it has been theoretically confirmed that the probability distribution of the new mathematical model of random noise is asymptotically normal with mean zero and Variance NC_0^2/2 for large N as shown in Eq. (16). Since the use of digital computer is inevitable for the experimental confirmation in consideration of the complexity of the problem, the new white noise model was experimentally simulated by a digital computer as in Figs. 1(a) & 2, and then we made use of chi-square test (cf. Table 2), autocorrelation functions (cf. Fig. 4), scatter diagrams (cf. Figs. 5, 6 & 7) and distribution curve (cf. Fig. 3) to see whether it met some important requirements of white noise. It seems that the contents described in this paper must also be applicable to the other fields of measurement on random phsical phenomena owing to its fundamental character of considerations.