The ray angle diagram in ocean acoustic propagation

The seldom mentioned ray angle diagram (RAD) allows visualization of important properties of acoustic propagation. Based on the sound speed profile and Snell’s law, the RAD presents depth on the ordinate and tan θ(z; cn) on the abscissa for the full ray cycle of selected rays, parameterized by the ray parameter, cn. The angle θ(z; cn) is the angle the ray makes with the horizontal at depth z. Equivalently, it is the angle of tilt of the wavefront at depth z. Because many useful characteristics of acoustic propagation depend on tan θ(z; cn), it is used as the abscissa of the RAD. The tilt integral is defined as the integral of tan θ(z; cn) with respect to z between the upper and lower limits of the ray. The tilt integral can be visualized by an “area” on the RAD. The WKB phase integral, which relates rays and normal modes, is an integral of tan θ(z; cn) with respect to depth. Individual mode characteristics and number of propagating normal modes are represented on the RAD. Relationships to the adiabatic invariant, time delay for matched field processing, depth dependence of ambient noise, range-averaged transmission loss, and low frequency mode cut-off are discussed.