## Cilindrical symmetry (two-dimensional case)

In a system with cilindrical symmetry (see Fig.2.1) the Eikonal equation follows directly from Eq.(2.18) and Eq.(2.19), by replacing with and nullifying any derivative along :
 (2.21)

under this conditions the system Eq.(2.16) becomes
 (2.22)

In Eq.(2.22) e represent, respectively, the horizontal and vertical components of sound slowness:

The differential of travel time can be written as

 Figure 2.1: Ray slope, , ray step and horizontal and vertical steps and for the case of cilindrical symmetry; the parameters are related through the following relationships:

The system of equations given by Eq.(2.21) and Eq.(2.22) can be rewritten also in a more classical fashion as[3]

 (2.23)

or, more compactly, in vector form:
 (2.24)

When sound speed depends on depth only the horizontal slowness is preserved:

this, in combination with flat boundaries, allows to infer the classical form of Snell's law along the ray:
 (2.25)

The set of initial conditions required to solve the two-dimensional form of the Eikonal equation is given by

where represents the launching angle, stands for the source position and is the sound speed at the source position.

Orlando Camargo Rodríguez 2012-06-21