318 



SHALLOW-WATER REVERBERATION 



correction is questionable at large angles, as has been 

 pointed out previously. In addition, the ship pitches 

 and rolls; at large angles even a small change in 

 orientation of the projector may make a large dif- 

 ference in the received reverberation. There is the 

 further complication that measured beam patterns 

 in the vertical plane have not always agreed with 

 the measured patterns in calibrating stations.^ These 

 arguments, taken with the overcorrections noted in 

 reference 2, suggest that a correction of 10 db or 

 more is about the maximum which can be safely 

 applied to measured bottom-reverberation levels, if 

 accurate values of the bottom-scattering coefficients 

 are to be expected. We can conclude that the use of 

 equation (41) of Chapter 12 to obtain m" for rays 

 leaving the projector at large angles (greater than 

 6 degrees) is quite questionable. 



Also, little information about the angular depend- 

 ence of m" could be obtained from rays leaving the 

 projector at angles within the main beam, that is, 

 with initial angles less than 6 degrees. For, in these 

 experiments the incident angle at the bottom of the 

 rays within the 6-degree cone was essentially con- 

 stant; at best, this grazing angle varied only slowly 

 with range. This fact alone would make fruitless any 

 attempt to determine, from the data of reference 2, 

 a detailed degree-by-degree dependence of m" on 

 grazing angle. Furthermore, the value of the scat- 

 tering coefficient itself, at ranges beyond the region 

 where the main beam strikes the bottom, becomes 

 more and more doubtful as the range increases, be- 

 cause of uncertainty in the value of the transmission 

 anomaly. These remarks explain why the data of 

 reference 2 were capable of giving m" accurately only 

 for the angle of incidence on the bottom correspond- 

 ing to the peak of the reverberation. 



It is worth noting that the virtual constancy of the 

 angle of incidence on the bottom, for rays within the 

 6-degree cone, should be a rather general result with 

 all types of refraction patterns. This conclusion is 

 deduced from Snell's law of refraction, as follows. 



Snell's law, which was proved in Chapter 2, tells 

 us that 



cos d = ~ cos ^0 



Co 



(4) 



where d is the bottom grazing angle, da is the angle of 

 the ray at the projector, c is the velocity of sound at 

 the bottom, and Co is the velocity of sound at the 

 projector. It is clear from equation (4) that the bot- 

 tom grazing angle will be smallest for the ray which 



leaves the projector at degrees. Thus, the derivative 

 dd/ddii equals zero at 9o = 0; and d necessarily varies 

 but little for all the rays leaving the projector within 

 a few degrees of the projector axis. 



15.3.2 Dependence on Frequency 



A report by UCDWR'' presents measurements de- 

 signed to determine the dependence of the bottom- 

 scattering coefficient on frequency. These measure- 

 ments were made at 10, 20, 40, and 80 kc, with the 

 transducers directed downward at an angle of 30 de- 

 grees with the horizontal. The measurements were 

 made in two shallow water areas near San Diego, both 

 with rocky bottoms. The ping lengths used were be- 

 tween 4 and 8 msec. Further details concerning the 

 bottom character and the experimental procedures 

 are given in reference 6. From comparison of the 

 measured reverberation levels with equation (54) of 

 Chapter 12, values of 10 log ni" were determined at 

 each frequency and at each of the two positions where 

 measurements were made. These values of 10 log vi" 

 were obtained assuming the transmission anomaly A 

 in equation (54) as zero; because measurements were 

 performed in very shallow water, this assumption 

 should introduce very little error. The results ob- 

 tained in reference 6 are tabulated in Table 3. 



An irregular variation of 10 log m" with frequency 

 is noted in Table 3, but according to reference 6 this 



T.\BLE 3. Backward scattering coefficients (10 log m") as 

 a function of frequency at 30-degree grazing angle. 



Frequency in kc 



10 



20 



40 



80 



Area I 

 Area II 



-11 

 -22 



-6 

 -17 



-21 



-14 

 -15 



variation is less than the estimated error of calibra- 

 tion. Also, according to reference 6, change in trans- 

 ducer patterns due to changes in frequency and 

 swinging of the ship at anchor could have introduced 

 errors compared with which the observed variation is 

 not significant. Thus there is no evidence that the 

 bottom scattering coefficient for rocky bottoms at a 

 grazing angle of 30 degrees has any systematic fre- 

 quency dependence for the frequency range 10 to 

 80 kc. The mean value of 10 log m", averaged for 

 10, 20, 40, and 80 kc is -10 + 3 db for position I 

 and — 19 + 3 db for position II. 



The mean values of 10 log m" at a grazing angle 

 of 30 degrees, quoted in the preceding paragraph, 

 should be directly comparable with the 30-degree 



