228 



TRANSMISSION OF EXPLOSIVE SOUND IN THE SEA 



sound propagated in a single homogeneous medium, 

 the expression (26) would simply equal the phase 

 velocity. For the disturbances we are considering 

 here, however, Xf is not independent of frequency, as 

 a glance at Figure 27 will show. 



The importance of the result (26) in shallow-water 

 transmission is that it enables us to understand the 

 dispersion phenomena in the ground and water 

 waves. The initial disturbance can be represented as 

 a superposition of normal modes having a very wide 

 range of frequencies. However, since the group 

 velocity is different for different frequencies, the dif- 

 ferent frequencies in this superposition will get sepa- 



FREOUENCY IN C 

 2500 



Figure 29. Dispersion in group velocities of normal 

 modes. Frequency in /; depth of water, h; velocity 

 of sound in water, c. 



rated out somewhat at long ranges, and each band 

 of normal modes of a given order and a given narrow 

 range of frequencies will be propagated with its own 

 group velocity. This effect is shown quantitatively 

 for a typical set of conditions in Figure 29. The curves 

 of this figure are derived from those of Figure 27 by 

 differentiation. Note that the group velocity varies 

 from the ground velocity Ci at the low cutoff fre- 

 quency to the water velocity c at very high fre- 

 quencies, but has a minimum at an intermediate 

 frequency. The existence of this minimum produces 

 an interesting effect, which will be described later. 



The main features of the disturbance received at a 

 distance from an explosive source can be explained 

 most simply by concentrating attention on one of 

 these curves, say that for the first mode. This will not 

 only be illustrative of the main characteristics shared 

 by_]all the normal modes, but will in fact provide a 

 rough prediction of what some of the actual records 

 to be discussed in Section 9.4.3 should look like. For 

 it has been pointed out that there exists a minimum 



frequency for each normal mode and that the fre- 

 quency for the first mode is the lowest. Thus if the 

 disturbance produced by an explosion is received with 

 equipment responsive only to sufficiently low fre- 

 quencies, the resulting signal can be interpreted in 

 terms of the first-order modes alone. Even when high- 

 fidehty recording equipment is used, the first mode 

 should dominate the initial or ground wave portion 

 of the disturbance, since it can be shown theoretically 

 that the amplitude of the first mode is greater than 

 the amplitude of higher modes in this region. ^^ 



Let us therefore suppose that we have a source of 

 sound which generates a transient disturbance con- 

 sisting entirely of a superposition of first-mode vibra- 

 tions of various frequencies. Since according to 

 Figure 29, the highest group velocity occurs for the 

 lowest frequencies above the cutoff, the first sound to 

 arrive at a distant hydrophone will be a wave train 

 whose frequency corresponds very nearly to point A 

 of Figure 29. The disturbance arriving a little later 

 will consist of frequencies having a slightly slower 

 group velocity, that is, of slightly higher frequencies. 

 Thus, the frequency of the received disturbance will 

 gradually increase with time until the value corre- 

 sponding to point B is reached. At this moment, the 

 very highest frequencies present in the original dis- 

 turbance start to come in, traveling in the Umit with 

 the velocity c. From this time onward, the received 

 disturbance consists of a low-frequency part and a 

 high-frequency part superposed, the former con- 

 tinuously increasing its frequency along the branch 

 BC of the dispersion curve, and the latter continu- 

 ously decreasing its frequency along the branch FG. 

 Eventually these two coalesce, and the disturbance 

 dies out at an intermediate frequency. 



All these characteristics are apparent in the theo- 

 retical pressure-time curve of Figure 30, which shows 

 the contribution of the first mode to the disturbance 

 produced under a typical set of conditions by a source 

 which emits a single positive-pressure pulse of short 

 duration. The portions of the curve corresponding 

 to the points A, B, C, F,G of Figure 29 are labeled 

 with these letters. Similar curves showing the con- 

 tributions of normal modes of higher order are given 

 in reference 23. These have lower ampUtudes than 

 that for the first mode, especially during the "ground 

 wave" phase, that is, the portion of the disturbance 

 which has traveled with a velocity greater than c and 

 thus hes to the left of B and F. According to the 

 present theory, which idealizes the bottom as a 

 homogeneous fluid, the variation of the pressure at 



