TRANSMISSION THROUGH A THERMOCLINE 



113 



TEMPERATURE 



SOURCE 



FiGiRE 26. Cause of layer effect. 



Layer effect was examined theoretically, from the 

 standpoint of ray acoustics, under "Combination of 

 Linear Gradients" in Section 3.4.2. The approximate 

 formula obtained in Chapter 3 was 



A = 10 log 



(11) 



In this formula, A is the transmission anomaly, hi is 

 the vertical distance between sound source and top 

 of the thermocline, 6h is the angle of inclination of the 

 sound path at the hydrophone, R the range, and ^o is 

 the angle of inclination of the sound path at the sound 

 source, as in Figure 27. To facilitate comparison with 

 experiment, this expression can be further simplified. 

 If the upward bending caused by the increasing pres- 

 sure is unimportant, which will be the case at ranges 

 which are not too long, then to an adequate approxi- 

 mation, do may be replaced by hi/R. If dk is calculated 

 by means of equation (81) of Chapter 3, equation (11) 

 then becomes 



A = 10 log ^ 1 + 



f^ 



2Acfl2 

 Cohl 



^ , / 2Aci?A 



(12) 



In most situations, the thermocline is not confined 

 to a thin layer but is a hundred feet or more in thick- 

 ness. Extensive intensity computations for simple 

 types of bathythermograph slides are included in a 

 report by WHOI.^' Some of these have been repro- 

 duced in Figure 25 of Chapter 3. It is evident from a 

 study of these figures that most of the drop in inten- 

 sity takes place in the top part of the gradient. With 

 increasing hydrophone depth in the thermocline, the 

 increase of total temperature change begins to be off- 

 set by the larger inclination with which the sound ray 

 enters the thermocline on its way to the hydrophone. 

 More detailed theoretical calculations show that the 

 temperature gradient in approximately the top Z)/3 ft 

 of the thermocline should be important, where D is 



HYDROPHONE 



Figure 27. Diagram clarifying layer effect formula. 



the depth to the top of the thermocline, sometimes 

 called layer depth. Since the exact choice of depth 

 interval should not be very critical, it has been cus- 

 tomary to use the temperature change AT in the top 

 30 ft of the temperature gradient in computations of 

 the intensity change to be expected theoretically. 



When there are several temperature gradients 

 present, or when the sharpness of the gradient in- 

 creases with depth, the theoretical intensity depends 

 in a complicated way on the temperature pattern. 

 However, an empirical study of the numerical in- 

 tensity computations summarized in reference 31 

 shows that the following procedure usually gives a 

 moderately good approximation to the theoretical 

 intensity found by ray tracing methods. Consider 

 separately each 30-ft interval of the thermocline. 

 Take the largest value of A found from equation (12) ; 

 this then gives the theoretical intensity for a given 

 initial ray inclination 6 when the ray reaches a depth 

 about 4/3 of the depth to the top of this 30-ft interval. 

 This computed intensity also applies in theory to 

 somewhat greater depths, since especially for deep 

 thermoclines the intensity increases only relatively 

 slowly with increasing depth below the depth of 

 minimum intensity. 



Only that part of attenuation which is due to the 

 sharp refraction at the top of the thermocline is taken 

 into account in the expression (12) for the transmis- 

 sion anomaly to points below the layer. If we assume 

 that the attenuation due to absorption is 4 db per 

 kyd, which is a reasonable estimate for 24-kc sound 

 in an isothermal layer, the formula (12) is replaced 

 by the following more realistic formula 



A = 0.004ft -I- 5 log I 1 -I- 



/ 2Aci?A 



(13) 



In formula (13), i2 is the horizontal range to the point 

 where A is measured, Ac is the temperature decrease 

 in the top 30 ft of the thermocline, and hi is the height 

 of the sound projector above the top of the thermo- 



