[ ASP Lab. TOP ]

When we clap our hands once between parallel, hard walls, we hear a sound called a "fluttering echo". A single handclap (i.e., an impulsive sound) is reflected by the walls repeatedly, and a train of pulses with periodic intervals is generated. This pulse train causes a specific sound sensation; that is, a fluttering echo whose frequency is a constant. In the fluttering echo, reflected sounds travel forward and backward in a one-dimensional pattern between parallel hard walls. But what happens when we clap hands in a three-dimensional reflective space? The frequency of the reflected sounds rises linearly with time. This phenomenon is not known widely. The reflected sounds were called "sweeping echoes" by the researcher who found. The generation mechanism is explained below.

Sweeping echoes were unexpectedly found by the researcher when an impulsive sound was generated in a relatively large regularly shaped room with three-dimensional highly reflective surfaces; i.e., walls, ceiling, and floor shown in Figure 1 (11m x 8.8m x 6.6m(height)). The researcher heard a strange sound, quite unlike the reverberation of a footstep, when he walked in the reverberation room. When he clapped his hands once, the frequency of the sound rose clearly. An analysis revealed that the frequency of the reflected sounds rose linearly with time. Figure 2 shows its spectrogram where the horizontal axis represents time and the vertical axis represents frequency, respectively. You can see the first sweeping sound rising up to about 1.5 kHz during the first 0.4 s and several subsequent sweeping sounds rising relatively slowly. Since the energy of the former is relatively high, it is called the main sweeping echo, and the latter are called sub-sweeping echoes. The generation mechanism of these sweeping echoes can be explained by using number theory, where the room is assumed to be cubic for simplification.

To hear the sound shown in Figure 2, click [sound].

[sound]

Figure 3 shows the mirror image sources generated in a cubic room based on geometrical acoustics. The figure shows a top view, and the size of the room is denoted by L wherein the source and the reception point are assumed to be located at the center of the room. When a pulse sound is generated at the center of the room, the arrival times and amplitudes of the observed reflected sounds are the same as those of the sounds that would be generated from the image sources shown in Figure 3. In other words, the reflected sounds are treated as sounds from the image sources. The coordinate origin O is set at the center of the room. Then, the location of each image source is represented by (n_xL, n_yL, n_zL) where n_x, n_y, n_z are positive and negative integers. The distance d between the origin and an image source of (n_xL, n_yL, n_zL) is represented by

The arrival time of the sound from the image source is obtained by dividing d by the velocity of sound c, as in the following equation:

Since these forbidden numbers account for 1/6 of all positive integers, we disregard the forbidden numbers and assume that M includes approximately all positive integers. Then, equation (3) indicates that reflected sounds (pulse sounds) exist at (L/c)² , 2(L/c)² , 3(L/c)² , ...; that is, they exist at equal intervals of (L/c)² on the squared-time axis.

A periodic pulse series has a fundamental frequency represented by the reciprocal of its interval. Therefore, when the interval of pulses is represented by equation (7), the fundamental frequency of the pulses at time t_m is expressed by

As described above, a pulse series from the image sources of a cubic room does not have completely equal intervals on the squared time axis because of the forbidden numbers. The influence of the forbidden numbers can be explained as the addition of an FNPT with negative amplitude, where an FNPT is a forbidden numbers pulse train, which has pulses corresponding to forbidden numbers on the squared-time axis. Figure 4 conceptually illustrates this phenomenon. Figure 4(a) shows the pulse series on the time axis of a cubic room for equal amplitudes. Some pulses are missing because of the forbidden numbers. These gaps are considered to be generated by adding an FNPT with a negative amplitude of the same value (Fig. 4(c)) to a pulse series with completely equal intervals on the squared-time axis (Fig. 4(b)). In other words, actual room echoes are composed of pulse train of Fig. 4(b) causing the main sweeping echo and FNPT (Fig.4(c)).

As shown in Eq. (5), the forbidden numbers are represented as 4^k(8m+7) (k,m = 0, 1, 2, ...). For a typical example, corresponding to k = 0 and m = 0, 1, 2, ... , the period of the FNPT is 8(L/c)²．Since it is 8 times of the period (L/c)² for the main sweeping echo, its frequency rises more slowly, at 1/8 the speed than that of the main sweeping echo. In the case of k = 1, 2, ..., the periods get longer and the frequencies rise more slowly. Such influence of the forbidden numbers causes the sub-sweeping echo.

[1] M. R. Schroeder, Number Theory in Science and Communication, (Springer-Verlag, Berlin) 
1984, p. 98.
[2] K. Kiyohara, K. Furuya, and Y. Kaneda, "Mechanism of the low-speed sweep sound perceived 
in a regular-shaped reverberation room ", J. Acoust. Soc. Jpn. (E) 21, 4 (2000) pp. 233-235.