The Eikonal equations

For a system with cylindrical symmetry the Eikonal equations can be written as [1]

$$\begin{array}{cccccc} \displaystyle { \frac{dr}{ds} } & = & ... ...\frac {1}{c^2} \frac{\partial{c}}{\partial{z}} } , \end{array}$$ (1)

where $r(s)$ and $z(s)$ represent the ray coordinates in cylindrical coordinates and $s$ is the arclenght along the ray; the pair $c(s)\left[ \xi(s),\zeta(s) \right]$ represents the tangent versor along the ray. Snell's law

$$\frac {\cos\theta(s)}{c(s)} = \makebox{ constant }$$ (2)

follows from the second equation in the case of ray propagation with a sound speed profile ($c = c(z)$) and flat waveguide boundaries.

Initial conditions for $r(s)$, $z(s)$, $\xi(s)$ and $\zeta(s)$ are

$$r(0) = r_s \hskip3mm , \hskip3mm z(0) = z_s \hskip3mm , \h... ... , \hskip3mm \zeta(0) = \frac {\sin\theta_s}{c_s} \hskip3mm ,$$

where $\theta_s$ represents the launching angle, $(r_s,z_s)$ is the source position, and $c_s$ is the sound speed at the source position. The coordinates are sufficient to obtain the ray travel time:

$$\tau = \displaystyle{\int\limits_{\Gamma}^{}} \frac {ds}{c(s)} ,$$ (3)

which is calculated along the curve $\left[ r(s),z(s) \right]$.