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

422 15. Underwater AcousticsFigure 15.6. Diagram used for derivation of the relation between gradient G and the radiusof curvature R of a sound ray.(e.g. 0.1% over a 100 m depth). But variations in speed due to temperature changesare much greater and can fluctuate wildly, particularly near the surface.When sound varies with ocean depth, the path of a ray through the medium canbe determined by applying Snell’s law (sin φ/c = constant). Because the rays ofthe greatest interest in the study of oceans are nearly horizontal, it is more usualto restate Snell’s law as follows:cos θ= 1 (15.7)c c 0where θ is the angle of refractive deflection made with the horizontal at a depthwhere the speed of sound is c and c 0 is the speed at a depth (real or extrapolated)where the ray would become horizontal.A complex profile of the propagation velocity versus depth such as that inFigure 15.1 can be simplified for analytical purposes by separating the profile intosmall enough segments so that the velocity gradient may be considered constantover its length. Advantage is taken of the fact that the path of a sound ray througha stratum of water over which the sound speed gradient G is a constant constitutesan arc of a circle whose center lies at a depth where sound speed extrapolates tozero.In Figure 15.6 we consider a portion of the ray path with a radius of curvatureR. It follows that z = R(cos θ 1 – cos θ 2 ), and the gradient G isG = c 2 − c 1. (15.8)zWe can combine the last two Equations with Snell’s law of Equation (15.7) whichnow yieldsR =− c 0G =− cG cos θ . (15.9)