In general, the rays are bent away from areas of higher sound speed.  This effect leads to corridors in which the waves are unable to escape.  I consider three novel cases.  The first is the shallow water corridor.

Fig 3.  Ray trace diagram for a source in the shallow water corridor.

In the figure above, the solid black line represents the ocean floor and the wavy cyan line represents the ocean surface (the surface was taken to be uniformly at 1000 m in the integration but I made it slightly wavy in the diagram to differentiate it from the horizontal ray trajectory).  The sound source is located along the left margin at 75 m depth.  The take off angles for the rays vary from 0^o (blue horizontal line) to -4^o (yellow curve) in increments of 1^o.  Three distinct classes of rays arise.  We have rays that are strictly refracted for take off angles of up to -2^o, a ray that is refracted-surface reflected (magenta curve), and a ray that is surface-bottom reflected.  Evidently, any rays with take off angles greater than -4^o will be surface-bottom reflected.  It’s important to note that the range axis scale numbers correspond to kilometers, while the depth is in meters.  It follows that the refraction occurs only over take off angles that are near to horizontal.  As the take off angle increases, the effects of the varying sound speed become less apparent.

            In addition to the shallow water corridor, the deep water may behave like a waveguide as well. 

Fig 4. Ray trace diagram for a source in the deep water corridor.

In fig 4. the source is again at the left margin but is now at a depth of 850 m.  The take off angle is relatively steep at 5^o, yet it corresponds to a refracted-bottom reflected trajectory.  In general, the deep water corridor is found to refract much steeper trajectories than the shallow water corridor will support.  However, no refraction will occur near the ocean floor where the sound speed profile is nearly constant.  Therefore, there is only one class of ray trajectories that remains in the deep water corridor, as pictured above.

            A case that is of particular interest to some acousticians is that of sound waves which become trapped in the thermocline itself and propagate over great distances without oscillatory behavior.  Consider the plot below.

Fig 5.  Ray trace diagram for an intermediate source point.

In fig. 5 the source is located at a depth of 450 m.  The take off angle has then been chosen such that the wavefront travels to the thermocline and is refracted to almost horizontal propagation, with the particular choice of angle maximized by trial and  given by theta = 2.837434796^o.  Here the ray trajectory is almost horizontal for over four kilometers.  Since the sound speed varies only in the z direction, if the trajectory ever becomes exactly horizontal it remains so throughout the trajectory.  However, the situation is an unstable equilibrium; any deviation from horizontal gives rise to exponential growth or decay for the trajectory as a function of z.  Indeed, just observe that if the horizontal component is not identically 0, then in (17) we have:

                        (21)

Which demands growth or decay for the z component of the ray trajectory.  Therefore, machine error alone will make it impossible for this trajectory to remain exactly horizontal indefinitely along a semi-infinite arc length.  However, for a point source wherein all possible take off angles are continuously exhibited there must be some take off angle that corresponds to horizontal travel along the thermocline.  This case is particularly interesting because a directed wave may travel along this waveguide with very little loss over exceedingly long distances.