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A wave-theoretical interpretation is given of pressure waves generated in shallow water by explosions of charges of T.N.T. ranging from 0.5 to 300 lbs., and recorded by Ewing and Worzel. (See accompanying paper, Explosion sounds in shallow water.)

The normal mode theory of propagation of sound in layered media, which was developed by the writer in 1941, was extended to cover the case of explosive sound, and the predictions of the theory about the shape and variation of amplitude in the received pressure pulse were investigated in detail. It was found that the theory predicted the existence of a series of readily identifiable new features in the pressure wave, each of which is characteristic of the depth of water and the structure of the bottom. A study of the original records, some of which are reproduced on Plates 1-11, revealed the presence of all the predicted phases. The characteristics of these phases were then measured, and the data were interpreted in terms of the structure of the bottom at the various stations. The deductions about the distribution of sound velocity in the bottoms, based on an analysis of the various features of the pressure waves, are given in Table A, and it will be seen that they agree among themselves.

The following results were obtained:

(1) A study was made of the dominant periods in the ground waves which are propagated along the various interfaces in the layered bottom, in order to verify the theoretical prediction that the deeper the interface (higher sound velocity) the longer should be the periods. A verification of this theoretical prediction is well illustrated in Figures 1 and 2, and to a lesser extent in Figure 3.

(2) An extensive investigation, covering an analysis of more than 40 records, was made of the dispersion in the water wave (which is illustrated by the third trace from the bottom on Plate 11). A technique was developed for determining from the records the speed with which each frequency in the water wave is propagated. The discovery made empirically by Ewing that this speed is a function of frequency only {see accompanying paper, Explosion sounds in shallow water) and is independent of the range was confirmed in all the records, as is shown in Figures 6-19. The shape of the mean dispersion curve at each station was successfully interpreted by an application of the normal mode theory in a layered liquid half-space. Theoretical dispersion curves form the background in Figures 6-19, and, with the aid of these, deductions were made about the sound-velocity distribution in the top layers of the bottom. The conclusions are given in columns 6 and 7 of Table A and in Table 1.

(3) The theory of normal modes was developed by the writer to a stage which enables one to compute the actual curve of pressure variation, as recorded by various types of receivers, due to an arbitrary explosion. A sample of such a theoretical pressure wave is shown in Figures 24A, 24, and 25.

(4) The following new features of the pressure waves were predicted by the theory of normal modes of a layered liquid half-space and were subsequently discovered and analyzed by the writer:

A) In case of a uniform bottom extending down to a depth many times the depth of water, the ground wave should begin with a so-called limiting period which is characteristic of the depth of water and the sound velocity in the bottom. The limiting period was identified and measured in the records taken at the Solomons Shoal station where the bottom is known to meet the requirement stated above, and the results are shown in Table 2. The value of 1.29 for c2/c1 obtained from the average observed limiting period, where C1 and c2 denote the sound velocities in the water and in the bottom, is slightly higher than the values deduced from the other features quoted in Table A, but this small discrepancy can be explained by the effect of the deep layers.

B) The water wave should arrive riding on a low-frequency wave called the rider wave; the frequency of the rider wave just prior to the arrival of the water wave is determined by the depth of water and the distribution of sound velocity in the bottom. The rider wave was identified and its period measured on all records taken at Solonons Shoal, Jacksonville Shoal, and Jacksonville Deep. The results are set out in Tables 3-5, and the resulting conclusions about the sound velocity in the bottom are quoted in Table A. Some illustrations of the rider waves can be seen in the records reproduced on Plates 1-9.

C) The amplitude of the water wave should increase with time to a maximum value and should decrease thereafter, while the period should remain constant after the maximum is passed. The value of this period, which will be referred to as the Airy period, is again characteristic of the depth of water and the structure of the bottom. Values of the Airy period are given in Tables 2, 3, 5, and 6, and the interpretation of the average values is given in Table A.

D) A three-layered medium in which the thickness of the intermediate layer is only of the order of the depth of water should possess dispersion characteristics similar to those of a medium with a uniform bottom. The existence of the intermediate layer should therefore not be revealed by a secondary arrival. Theory also predicts that the amplitude of the rider wave should be relatively low in such a medium (by a factor of 1/5 to 1/10), while the water wave should be of normal intensity. The stations of Virgin Islands Shoal and Virgin Islands Deep which, judged by the combined evidence from the refraction data and the dispersion data in the water wave, have a veneer of mud of a thickness of the order of the depth of water covering a high-speed coral base, would be expected to fall into this class. The records taken at these stations were found to be lacking in secondary arrivals and to be devoid of rider waves, as is illustrated in Plates 8 and 9. The success of the theory in explaining the appearance of the records taken at the Virgin Islands, which were entirely different from the records taken at all the other stations, is very encouraging.

(5) Theoretically the maximum amplitude in the water wave should vary like the inverse 5/6-th power of the range, whereas the observations of Ewing and Worzel indicate that in some stations the maximum amplitude varies like the inverse square of the range. We have, of course, neglected absorption and scattering, but, as I have already suggested, it would be interesting to check the experimental determination of variation of intensity with range.

(6) Our study shows that in all stations the speed of sound in the first 30 feet of the bottom is no more than about 10 per cent greater than in water. This result conforms with Ewing's finding that all bottom samples were muddy.

(7) A complete theory of propagation of sound, both of single-frequency and of the explosive type, in layered media is developed in Part II of this paper. This includes a discussion of the "ray theory" and the wave theory. One interesting theoretical result is that in case of a density discontinuity at the bottom the normal modes are not orthogonal, nor is their amplitude, in case of a point source, correctly given by standard theory of normal modes.

Another of the new results arrived at is that, when the wave length of sound is of the order of the depth of water, the amplitude of the pressure should decrease at large ranges like the inverse square of the range, as in the Lloyd Mirror Effect. The asymptotic expressions given in Eqs. (32) and (33) are strikingly verified in Figure 23, in which they are compared with values obtained by numerical integration of the integral in the exact solution.

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