224 



TRANSMISSION OF EXPLOSIVE SOUND IN THE SEA 



in Figure 26A and labeled II^; many other such 

 paths, not shown, are also possible; some of them in- 

 volving additional reflections from the water-bottom 

 interface as was the case for I'g. This second group of 

 arrivals will in turn be followed by another group, 

 exemplified by III„ in Figure 26A, involving two re- 

 flections from the free surface, and so on. Mixed in 

 with these arrivals along paths which enter and leave 

 the bottom will be those along paths lying wholly 

 in the water, shown in Figure 26B. These paths have 

 for simplicity been drawn for the case where the 

 velocity of sound in the water is uniform. In practice 

 most of the experiments performed so far have en- 

 countered isothermal water with consequent upward 

 refraction; this case, which will be discussed later, is 

 in most respects little different from the uniform case 

 considered here. The first arrival among these water 

 rays will be along the direct path I„, the second along 

 the surface-reflected path II„, the third along a path 

 III„ involving one reflection from the bottom, and 

 so on. 



Thus if the predictions of ray acoustics were valid, 

 we should expect the signal received at the hydro- 

 phone to consist of a number of evenly spaced groups 

 of pulses of diminishing strength, corresponding to 

 the "groimd rays" shown in Figure 26 A, plus a 

 number of individual pulses starting at a later time 

 and separated by graduaUy increasing intervals, 

 which correspond to the "water rays" shown in 

 Figure 26B. Of these various arrivals, some are posi- 

 tive pressure pulses, others negative, according to 

 the number of phase-changing reflections each has 

 suffered. 



The extent to which the predictions of ray theory 

 can be trusted in a case of this sort can be estimated 

 by resolving the explosive pulse into a superposition 

 of sine waves by use of Fourier's theorem, as de- 

 scribed in Section 9.2.4, and then applying the criteria 

 given at the end of Section 3.6.2 for applicability of 

 the eikonal equation to sinusoidal waves. It is clear 

 from these criteria that the condition for ray theory 

 to be applicable to a sine wave along a path of the 

 type Ig, llg, etc., in Figure 26A, is that the maximum 

 depth of the path, shown as di for ray Ig, should be 

 large compared to the wavelength of the sound in 

 the bottom. This condition will be fulfilled by the 

 highest frequencies in the Fourier resolution of the 

 explosive pulse, but not by the lowest frequencies; 

 moreover, the frequency above which ray theory is 

 apphcable recedes to higher and higher values for the 

 successive arrivals !„, 11,, III^, etc. As is to be ex- 



pected, this critical frequency approaches infinity as 

 the magnitude of the velocity gradient in the bottom 

 decreases to zero, since the depths of penetration dj, 

 etc., of the rays approach zero. 



A similar consideration of the disturbance propa- 

 gated through the water suggests that ray theory 

 should fail for frequencies of the order of c/h and 

 smaller, where h is the depth of the water. This hmit 

 has little meaning, since this frequency can be ex- 

 pected to be lower than the frequency at which the 

 ray picture fails for the ground rays, and we cannot 

 make a clear separation between the ground dis- 

 turbance and the water disturbance after we have 

 abandoned the ray concept. 



We may thus expect the pressure variation which 

 would be recorded by a very high-fidelity receiver at 

 H to consist of the succession of pulses which ray 

 theory would predict plus a correction which is made 

 up almost entirely of low frequencies. For the dis- 

 turbance due to the shock wave from the explosion, 

 the times of the various ray arrivals can, ideally at 

 least, be identified on the oscillogram of the received 

 pressure by the occurrence of sharp jumps in the 

 pressure; these jumps, due to the sudden rise at the 

 front of the shock wave, cannot easily be obliterated 

 by the low-frequency correction (see Figure 13). 

 Since, as explained above, the intensities of the ar- 

 rivals predicted by ray theory are zero for a imiform 

 or downward-refraction bottom and are small for a 

 bottom with weak upward refraction, we should not 

 be surprised to find the distiorbance received by the 

 hydrophone to be dominated by the low-frequency 

 portion, with only a few detectable traces of the ray 

 arrivals. 



To determine the nature of the low-frequency cor- 

 rection just mentioned, it is necessary to study solu- 

 tions of the wave equation similar to those considered 

 in Section 2.7.2. In a report prepared by CUDWR,^' 

 it is shown how the normal modes of vibration of 

 water and bottom can be computed and superposed 

 to correspond to the disturbance produced by ex- 

 plosive source. The mathematical details are too 

 complicated to be given here;"= however an attempt 

 will be made below to explain in a simplified manner 

 the physical basis for some of the most important 

 results of reference 23. In particular, it will be shown 

 how many characteristics of the signal received at the 



° The reader who wishes to study the mathematical theory 

 of normal modes will find it profitable to study also the treat- 

 ments devoted primarily to single-frequency sound in deep 

 water^ and electromagnetic waves in the atmosphere.* 



