In our acoustical environment there are various sound fields which are quite complex or essentially non-deterministic. In order to investigate the properties of these sound fields it is often convenient to consider their stochastic models composed of random wave elements. First we consider a stochastic sound field composed of many plane monochromatic wave elements. Rayleigh distribution is derived for probability density of pressure amplitude when the phases of plane wave elements are purely random. But this distribution is not necessarily dependent on the nature of wave number vector (i. e. direction of propagation) of each plane wave element. If we are concerned with diffusion of sound, therefore, it may be more rational to investigate the stochastic behaviors of particle velocity and flow of acoustic energy in the field, because they are also positively dependent on the direction of propagation of each component wave. Probability densities of them are derived when the wave number vector of each component wave is purely random in one-, two- and three-dimensions respectively. Especially the Maxwell distribution is obtained for the probability density of flow of acoustic energy in the three-dimensional case. These results are shown in Figs. 1 and 2. Covariance matrix for particle velocity at two points in the random sound field mentioned above is also derived and shown in Fig. 5. To express the random structure of the field, this tensor quantity may be more preferable than the usual correlation coefficient (i. e. scalar quantity) for sound pressure. The latter us obtained by simply averaging the former tensor elements. Then the flow of acoustic energy is investigated in a sound field emitted by a random distribution of acoustic point sources. Probability density of the flow is given by the Holtsmark distribution which is known well in stellar dynamics and shown in Fig. 6. Some remarks on a relationship between this and Maxwell distributions are also mentioned.
There is Rayleigh's expression for the sound field excited by a transmitter. However, since that is two-dimensional integral expression, it is difficult to solve it analytically expect special cases such as the sound pressure on the axis of a disk transmitter. Then, the nearfield of a disk transmitter was calculated numerically with a digital computer. It seems that it takes much time to get the result with good precision by the method. If the Rayleigh's expression is transformed into one-dimensional integral, it can be computed easily. One of one-dimensional integral expressions is presented here. This is the integral of the quantity, which is the function related to the wave form of transmitted signal multiplied by the ring function related to the shape and dimension of a transmitter, of the distance from an observing point to the area element of the transmitter. Therefore, it is useful for the numerical calculation of the sound pressure at an observing point due to various transmitted signals. The expression (1) is Rayleigh's equation. This is transformed into the expression (7), in which R(r) named as ring function is determined with the shape of vibrating surface and an observing point. In Fig. 2, the locus of the points to which distance is r from the observing point is a circle. R(r) is the rate of the arc length contained within the vibrating surface to whole circle as shown in Fig. 2. Fig. 6 is a flow chart for the calculation of the sound field of a disk transmitter. Fig. 7 shows the result of the calculation about a circular concave transmitter excited with continuous sinusoidal signal. In another example, a signal is applied to a disk transmitter so that the vibrating velocity of disk surface is as shown on the right bottom of Fig. 10. Then, Fig. 10 (1)-(6) show the pressure wave form at each point in Fig. 9.
A delay Line having a delay time of a few millisecond at low frequencies is useful in signal processing in sonar, voice and data communications. A compact mechanical delay line utilizing torsional waves propagating in a central wire can be constructed to meet these demands. Cross-bars are spot-welded to the central wire at regular intervals so that the propagating speed of the wave I greatly reduced (Fig. 1). The central wire acts as compliance and cross-bars act as inertia. Hence characteristics of the line can be represented by an equivalent low-pass filter circuit (Fig. 2). Design formulas are derived (Fig. 4). It is shown that the product of the delay time per unit section and the cut-off frequency is nearly constant (Fig. 3). The effect of anti-resonance of cross-bars on the delay characteristics is also discussed. Performance of the line at the base band and an intermediate frequency is presented. At the base band, a bimorph ceramic transducer is used at the input and an electrodynamic transducer is used at the output. Supression of spurious flexural waves is achieved by either push-pull excitation (Figs. 5, 7) or orthogonal arrangement of input and output transducers (Figs. 6, 8). A delay of 2 msec over 3kHz bandwidth is obtained in a line of 30 cm in length (Table 1, Figs. 9, 10). Supression of spurious reflection is found to be difficult. At an intermediate frequency of 36 kHz ceramic torsional transducers are used. A delay of 0. 8 msec over 4. 9 kHz bandwidth is obtained in a line of 30 cm in length (Fig. 14, 15, Table 2). The transducer consists of circular ceramic sectors bonded together to make a disk which is sandwiched between metal disks. Poling direction of sectors are so arranged that a circumferential poling can be approximated (Fig. 11). The effect of the number of sectors on the capacitance ratio is analyzed (Fig. 12). It is found that four sectors are good enough in practice.