# Solving the transport equation

The solution of the Eikonal equation allows to calculate the Jacobian , between the usual set of cartesian coordinates and a particular set of ray coordinates , where stands for the ray arclenght and stand for some sort of auxiliary ray angles. In general, represents the cross section of a ray tube propagating from the source to the receiver. Let us define as the unitary vector tangent to the ray:
 (2.26)

Therefore, the first term in the transport equation, combined with the Eikonal equation, can be rewritten as

As for the second term, let us notice that the analytical properties of the Jacobian[3] allow to write that

The previous expressions transform the transport equation into
 (2.27)

which can be easily integrated to provide the result

where stands for a constant, which depends of the type of acoustic source. The solution to the wave equation can then be written as

In the proximity of a point source one can identify the ray coordinates with the spherical coordinates , centered at the source position (see Fig.2.2); therefore, it can be written that

with standing for the launching angle. On the oher side, close to the source, the acoustic field can be approximated as a spherical wave, which implies that

it follows from this relationship that

The classical solution can then be written explicitely as

 (2.28)

Unfortunately, the classical solution based on the Jacobian suffers from a serious drawback. Each time a ray tube narrows into a single point the Jacobian becomes zero (the points at which are called caustics), and the acoustic field exhibits a singularity at such point. Removing such singularities from the solution can be achieved by substituting the classical solution with the solution based on Gaussian beams.

Orlando Camargo Rodríguez 2012-06-21