With equations (17) and (18) in hand, a direct integration to obtain the ray trajectories is readily available.  Only a profile of sound speed verses depth is necessary to specify all of the pertinent parameters.  I consider a relatively shallow, static oceanic region exhibiting a strong thermocline near the surface.  For relatively static oceanic thermoclines the physical properties are quite complicated and have been explored by the likes of such fluid dynamics superstars are Rossby and colleagues, but for a detailed yet accessible discription I refer the reader to Stern [8].  The idea is that when a large body is heated uniformly from above no convection occurs to disrupt the resulting stratification [2].  Thermoclines have been experimentally explored in great detail, with particular attention paid to their effects on oceanographic acoustics.  Realistically, the temperature profile in the ocean tends to be time varying even in regions of strong stratification [7].  Clearly, these fluctuations take place on a much slower time scale than acoustic propagations, so they are not considered.  A typical temperature profile in the neighborhood of a strong thermocline is presented below, alongside the profile used for the model.

Fig 1.  Experimentally realized tropical oceanic temperature profile compared to derived model [3].

 

Upon viewing many measured temperature profiles I felt that a good approximation could be obtained from a sigmoid curve.  Specifically, in the plot above, the temperature profile is given by:

The constants in the equation were found by varying the parameters until a “good” fit to the experimental data was found based on comparison to empirically created plots like the one above.  This profile for T has the added benefit of providing an analytic expression for dT/dz, which is required to compute dc/dz in the ray equations. 

            Given a temperature, salinity, and pressure profile for a body of water, generating a sound speed profile may still present a considerable challenge.  In fact, efforts to reconcile models of long range acoustic propagation speeds with experimentally verified data may still be underway [8].  Here, the formula developed by Mackenzie (1981) is utilized [8]:

                  (20)

where S is the salinity in parts per thousand (ppt) and z₀ is the depth of the ocean floor.  The model is considerably simplified by taking the salinity profile to be constant.  In fact, for many parts of the ocean the average salinity is very near 35 ppt [6].  Therefore, I take the salinity to be uniformly 35 ppt for all further considerations.  Now a sound speed profile is readily computed.

Fig 2.  Sound speed profile for shallow, constant salinity ocean water in the presence of thermal stratification.

 

One final consideration needs to be made for a complete ray trace model: the boundary conditions.  Again, I consider the simplest case of lossless reflection at the surface and ocean floor.  In this case, the boundary condition is that the angle of incidence is equal to the angle of reflection, which corresponds to a replacement of z with –z upon incidence with the boundary in equations (17).   This is typically an excellent approximation for the surface where the relative impedance difference in the atmosphere and the ocean prevents transmission [5].  However, for the ocean floor it assumes a very smooth, hard floor, which is usually not the case.  In the case of a silty ocean floor considerable transmission may occur, as well as ray displacement and other phenomenon [5].   The integration was performed using an explicit Runga-Kutta 2 [RK2] method with the local sound speed updated following each discrete RK2 step [4].