THE SCIENCE AND APPLICATIONS OF ACOUSTICS - H. H. Arnold ...

15.9 Mixed Layer 423When G is constant, and hence R is a constant, the path of the ray is therefore acircle. The center of curvature of the circle lies at the depth where θ = 90 ◦ whichcorresponds to c = 0. In the case illustrated in Figure 15.6, the speed gradient isnegative, so R is positive. Otherwise, if the speed gradient were to be positive, Rwould become negative and the path would refract upward.Once the radius of curvature of each segment of the path is established, theactual path can be traced through graphic or computerized means. Let the initialangle of deflection of the ray be designated θ 0 , and use be made of the geometryof Figure 15.6 along with Equation (15.9). The changes in range r and depth zare given byr = 1 Gz = 1 Gc 0(sin θ 1 − sin θ 2 )cos θ 0(15.10)c 1(sin θ 2 − sin θ 1 )cos θ 1(15.11)Applying the small angle approximations (cos θ ≈ 1, sin θ ≈ 0, etc.) and eliminatingθ from the last two Equations yields a convenient approximate relationshipbetween the range and depth increments along a ray for ∗θ∗ less than 20 ◦ and forr ≪|c 1 /G|:z = tan θ 0 r − G2c 0(r 2 ). (15.12)15.9 Mixed LayerWave action can cause the water to mix in the surface layer, thus creating what iscalled a mixed layer. The positive sound-speed gradient in this layer entraps soundnear the surface. After it is developed, the mixed layer tends to exist until the sunheats up the upper portion, decreasing the gradient. This heating effect engendersa negative gradient that leads to a downward refraction and the loss of sound fromthe layer. Because this occurs later during daytime, this effect became known asthe afternoon effect. During the night, surface cooling and wave mixing permitthis isothermal layer to reestablish itself.A computer-produced ray diagram (the discontinuous form of the rays are dueto the manner the velocity profile was subdivided in the computer program) fora source in a fairly typical mixed layer is shown in Figure 15.7. The conditionsfor which the diagram was plotted, the ray leaving a source at 1.76 ◦ becomeshorizontal at the base of the layer, Rays leaving the source at smaller angles stayentrapped in the layer; and rays that leave the source at greater angles are sent intothe lower depths of the sea. A shadow zone is created beneath the mixed layer ata range beyond the direct and near sound field. This zone is isonified by scattersound from the sea surface and by diffusion of sound out of the channel, causedby the nature of the lower boundary.