1973 年 29 巻 9 号 p. 509-516
The original infinite theory of Cremer doesn't coincide with experimental results, particularly on the directivity characteristic. By modifying the symbol for radiation angle (Fig. 1), we have Eq. (1) instead of Eq. (2), which means that the radiation angle is much more effective than expected before, and then, the condition of radiation affects the resultant transmission. We tried in this paper, therefore, to modify the infinite wall theory particularly on this feature. In a real wall with finite dimensions the transmitted wave diverges in all directions (Fig. 2), and its intensity I_t depends on the velocity of wall u_w in a certain functional form as I_t=f(u_w), u_w on the pressure of incident wave p_i as u_w=g(P_i) and finally the transmitted energy e_t is expressed by Eq. (4). By the aid of radiation factor λ(ka, kb) for a rectangular wall given by Eq. (12), calculated from the fundamental formula of Lord Rayleigh, Eq. (8), we can arrive at Eq. (11) as the functional form of e_t, and also by utilizing Eq. (3) for u_w from the original theory, we have practical forms of insulation, normalized SPL difference D'_n for single directional incidence and Sound Transmission Loss TL (Sound Reduction Index R in ISO R 140). Radiation factor, usually substituted by λ(ka) for a square wall, has two significant variables, incident angle θand (ka), k=w/c and S_w=(2a)^2 is the wall area. λ(ka) is illustrated in Fig. 4, where the dotted linef or an infinite wall expressed by Eq. (14) corresponds to the 'Abstrahl-Faktor' of Gosele. Curves for ordinary wall tail off from higher value of θ, when (ka) diminishes from infinity, and change completely to a flat characteristic given by Eq. (13), when (ka)<1. It remains rather immutable to θ in ordinary windows, because (ka) is not so large in the lower frequency region (Table 2). Fig. 5 shows also λ(ka) taking (ka) as abscissa, in which it is noticed that the transmission through wall is quite similar to a cone-action of loudspeaker, because (pc)×λ(ka, kb) expresses, refered to Eq. (15), the radiation resistance of vibrating rectangular-board per unit area. Direct application of the infinite wall theory to real walls would be equivalent, therefore, to the assumption that the diameter of cone-speaker would be always sufficiently larger than the wave-length of sound wave. D'_n and TL are defined by Eq. (16) and (20), leading to their theoretical expressions Eq. (19) and (24). Q in Eq. (24) is average of λ(ka, kb) in special directions of incident waves defined by Eq. (23) and calculated in Table 1 for the square wall, which gives the spatial average of D'_n as D'_na in Eq. (26). Figs. 6 and 7 show calculated values of D'_n and TL. Incident angle θhas generally few influence on D'_n, consequently the spatial average D'_na calculated from TL comes in good agreement with D'_n. TL becomes a little larger than the value for the Limp-Wall-Law of A. London, when S_w diminishes from infinity, but only with a slight deviation from it, except when S_w<1m_2. These tendencies in the experimental observations already known would indicate the importance of diffraction in the transmitted wave, and finally the fact that there is approximately no directivity in the behavior of windows would present a practical advantage to experimental works.