# The Gaussian beam approximation

The starting point for the approximation of the acoustic field as a Gaussian beam is given by the analytical expression
 (3.1)

where stands for sound speed along a ray trajectory:

and represents the normal to the ray, which can be considered as two-dimensional vector:

such that , where was already defined by Eq.(2.26). In what it follows the projection of sound slowness along will be written as .

While is a real vector both matrices and are complex. Therefore, the complex part of the product induces a Gaussian decay of ray amplitude along the normal (see Fig.3.1), while the real part introduces phase corrections to the travel time along . Additionally, a proper choice of initial conditions for will ensure that det , which frees the Gaussian beam approximation of singularities.

Both components of and can be considered as being dependent of a particular set of local ray parameters, let's say, ray arclength , plus angles and . At any point of the ray one can introduce a set of three orthogonal unit vectors, known as the polarization vectors; naturally, the first polarization vector is ; the other two polarization vectors, which are going to be represented as and , are within the plane perpendicular to (see Fig.3.2).

The vectors and define the possible orientations of and at any coordinate of the ray:

Besides matrices and the Gaussian beam approximation involves two more matrices, called and ; all four matrices are related through the following relationships:

 (3.2)

where
 (3.3)

 (3.4)

and
 (3.5)

where

The components of and correspond to partial derivatives of the ray normal and normal slowness, along the auxiliary parameters and [5]:

Orlando Camargo Rodríguez 2012-06-21