20 km 



60 km 



100 km 



250 km 



60 - 



80 



60 



to 80 

 O 



60 



2 

 < 



80 - 



60 - 



80 



— — -I- — — 1 [ — — n 1 1 — ■ — T ■ — I — — I n I 



I I I . I . J_ . 1 . . -L- ^ 1 



I I . I I _l_ . 1 . -l_ 1 



' I . _1_ I . . J ^ 1 1 1 



-10° 



-5° 



AXIS 

 RAY 



+5° 



50 



100 



50 100 50 100 



TIME — h 



50 



100 



FIG. 6. Transmission-loss time se- 

 ries at several ranges and depths. 

 Each section shows the intensity that 

 would have been observed by a single 

 hydrophone taking data once an hour 

 for 128 h. Each column represents a 

 particular range from the CW source. 

 Each row represents a particular ray 

 followed from the source. (Positive 

 angles correspond to downward rays 

 from the source.) That is, at each 

 range, the depth is chosen as the depth 

 that particular ray passes through at 

 that range. 



Is given by the canonical profile'" and the source is on 

 the sound axis (z^ = 1000 m). It is apparent that after a 

 few tens of kilometers the sound arriving at various 

 points has a complicated directional character due to 

 multiple paths. For example, at 60 km on the axis 

 sound should arrive from three well-separated direc- 

 tions. Note also that our absorbing bottom at 3.5-km 

 depth prevents any surface or bottom reflected energy 

 from propagating beyond about 20 km. 



It is possible that some of our results for acoustic 

 signals traveling through internal waves may be under- 

 stood in terms of internal-wave effects on individual 

 rays. It will be well to remember, however, that be- 

 yond the 20-km range a single hydrophone will in most 

 cases receive more than one ray from the source. 

 This multipath effect is crucial to understanding long- 

 range fluctuations. 



Figure 6 shows the computed transmission loss as a 

 function of time at several ranges for 100-Hz acoustic 

 signals traveling through the internal wavefield. The 

 point source is at a depth of 1000 m. Each row shows 

 results for a particular ray which has been followed 

 from the source by integrating Snell's law (e.g., the 

 hydrophone at 100-km range for the 6° ray is at the 

 depth corresponding to the 6° ray at that range). The 

 1-h time steps clearly undersample the fluctuations, 

 but the general character of the series is clear. We 

 see that internal wave sound-speed fluctuations cause 

 5-30-dB fluctuations in received intensity, comparable 

 in size to those observed in field experiments.' 



We have used a vertical beamformer (see Appendix) 

 to separate the different ray arrivals at various ranges 

 and depths. Figure 7 shows a time history of one 

 beamformer output. A single hydrophone would co- 

 herently add the many rays, each of which are seen in 

 Fig. 7 to vary in direction and intensity. If the peak of 

 the ray of interest is chosen at each time, then a time 

 series for the intensity of that ray can be plotted. 

 Figure 8 shows time series for the particular rays 

 corresonding to the single hydrophone results in Fig. 

 6. It is evident that the fluctuations of a single ray are 

 considerably muted compared with those of a single 

 hydrophone which is subjected to a coherent addition 

 of all rays. 



Figure 9 shows the rms intensity variation as a func- 

 tion of range for four rays. {[((lOlog/)^) - (101og/>^f '^ 

 is plotted.} Both single hydrophone and ray-peak re- 

 sults are plotted. The reduction in fluctuation that re- 

 sults from selecting a ray peak is clear. In addition, 

 the smoothness of the rms value as a function of range 

 for the ray peak gives grounds for hope that a simple 

 single-ray theory might be used to predict the range 

 dependence of the fluctuations. 



IV. SUMMARY AND CONCLUSIONS 



We have developed a parabolic-equation acoustic 

 propagation code which sends CW sound signals 

 through a time-dependent random internal wavefield 

 superimposed on a deterministic sound channel. ' The 

 output is the complex pressure field as a function of 



206 



