268 SHOCK WAVE MEASUREMENTS 



hundred feet, the effects may be very large for conditions encountered 

 in the ocean, and at intermediate distances they may cause measurable 

 effects in some investigations. A general discussion of the various 

 phenomena belongs in a treatise on underwater sound transmission. 

 The discussion here is intended merely to indicate the nature of the 

 effect by a simple example which is adequate to show the differences to 

 be expected for propagation over short ranges. 



The major factor in determining the velocity of sound in the ocean 

 at shallow depths is the temperature of the water, an increase of 10° F. 

 increasing its value by about 40 ft. /sec, the exact figure depending on 

 the temperature, depth and saline content. In the ocean, different 

 horizontal layers of water are not at the same temperature and the 

 result is a vertical gradient of sound velocity. The simplest case of 

 this kind, and the only one we shall consider, is that of a uniform nega- 

 tive velocity gradient, corresponding to warmer water at the surface. 

 If, as in Fig. 7.22, a wave is initiated at point 0, different parts of the 

 front travel at different rates, the upper portions advancing faster. As 

 a result, the wave front becomes distorted, and a sound ray drawn nor- 

 mal to any part of the front becomes bent downward as it progresses 

 (except for the part of the front travelling vertically) . This bending is 

 analogous to optical refraction, and the path of the part of the front 

 which reaches points P from is not the straight line joining but a 

 curved path. This path is the one of least time for a ray from to P, 

 and it is easily shown that Snell's law describes the inclination of suc- 

 cessive parts of the ray, just as in geometrical optics. 



If the negative gradient is a uniform one such that the velocity 

 Co{d) at a depth d is given by Co{d) = Co(l — Kd) where Co is the velocity 

 at the surface, the path between and any other point is a circular arc 

 joining the points with its center in a horizontal plane at a depth 1/K 

 below the surface, as indicated in Fig. 7.22. This sketch is exaggerated 

 to show the nature of the effect; for a temperature difference of 50° F. 

 between the surface and a depth of 1,000 feet, K = i X 10"^ ft.~^ and 

 the plane of centers is 25,000 feet below the surface. Even for this 

 rather extreme condition, the actual path between points and P a few 

 hundred feet apart is not much longer than the straight line (the dif- 

 ference at 1,000 feet is 0.07 feet in this case). The fractional change in 

 time of travel from that computed for a straight line is even smaller 

 than the path difference, as is qualitatively evident from the fact that 

 in time the curved path is shorter, despite its greater length in feet. 

 The amount of difference depends on the relative position of points 

 and P, as well as their separation, but the percentage change is, to a 

 first approximation, less than {KUy, where R is the separation. For 

 K = 4 -10"^ ftr\ R = 1,000 ft., the time difference is thus less than 

 0.04 per cent of the total time. The changes introduced by refraction 



