Derivation of the wave equation for travelling pressure waves in fluids is found in most graduate mathematical physics textbooks.  I follow Butkov in this derivation, with some details omitted for brevity[1].  Note that in text greek letters are spelled out explicitly due to limitations in the software

            Consider a volume element of an inviscid, irrotational fluid in which weight and body forces are negligible in comparison to the force associated with pressure.  Then the total force on the volume element satisfies:

                   (1)

Where the last equality follows from the divergence theorem.  Since V is arbitrary we have:

             (2)

It is physically clear,  (and is easily verified through dimensional analysis), that fluid flow velocity is negligible compared to the velocity of propagation of sound waves for oceanic flows.  The v ·grad v term in the total derivative may then be omitted to obtain:

              (3)

As always, the fluid must satisfy continuity conditions:

                     (4)

For compression waves in water the actual deviation from the local background water density has been shown to be exceedingly small, such that grad rho » 10^-11 gm/cm⁴ [1].  We are then justified in making the approximation:

                 (5)

                 (6)

Where rho₀ is the local background fluid density and is usually taken to be a constant for water independent of depth or other considerations.  Taking the divergence of (2) and the time derivative of (6) to remove v we have an equation in p and rho:

                  (7)

  These variables can be related through the observation that pressure and density changes are adiabatic in sound waves (negligible heat is exchanged between adjacent fluid elements):

                   (8)

Solving equation (8) for rho and twice differentiating yields:

Finally, then, defining c² = p₀g/rho₀ we arrive at:

               (10)

Which is, of course, just the wave equation in p.  In order to create a ray tracing algorithm to sketch the direction of travel of the disturbances we seek a harmonically varying solution for p with a source point located at X_s.  Such a solution must satisfy the Helmholtz equation (here given in Cartesian Coordinates):

              (11)

At this point I have followed the literature in development of the governing equation for propagation of sound waves.  Despite the linearizations and approximations made in deriving the Helmholtz equation it remains an excellent approximation for most sound waves.  Now, however, we run up against the fact that the refraction of the waves is frequency dependent, making a single comprehensive ray tracing algorithm exceedingly difficult.  It is customary, therefore, to derive ray tracing equations in the high frequency limit as follows (see Jensen et al for details) [5].  Seek a solution for p in the form of the series expansion:

                      (12)

Differentiating p yields:

                  (13)

Inserting this expression into the Helmholtz equation and equating like terms in omega we find that the highest order term is the order omega² term and it satisfies:

              (14)

At this point we approximate and discard all but the highest order terms.  This nonlinear PDE is referred to as the Eikonal equation [5].  While the function tau is as yet unknown, we have made progress because the level curves of tau represent the wavefronts of the harmonic waves (refer back to (12)).  Furthermore, we may note that the vector (1/c grad) tau is a unit vector that is perpendicular to the wavefronts.  This is exactly what we need to define the ray trajectories.  Introducing the arc length parameter s the ray trajectories must follow:

                  (15)

The final step is to utilize equation (14) to remove the unknown function tau from the equation.  This is accomplished through differentiating (15) with respect to s and grouping the right hand side into (tau)² terms.  This process is a tedious exercise in vector calculus, so here I only state the result:

               (16)

If we consider a system with azimuthal symmetry this equation can be re-written as a system of first order ODE’s:

                  (17)

The initial conditions for our system are specified by the position of the point source along with the take off angle of the specific ray we wish to follow:

                  (18)

These equations approximate the trajectory of energy travel for high frequency waves quite effectively.  However, in order to determine attenuation or low frequency effects more terms in the expansion for p must be considered [5].  While an in depth understanding of sound wave propagation requires detailed discussion of these and other considerations, for this project I terminate theoretical development here and base my models on equations (17) and (18).