# The acoustic wave equation

The starting point for the discussion of the ray tracing is given by the acoustic wave equation, which in the case of a watercolumn with a constant density can be written as
 (2.1)

where stands for the pressure of the acoustic wave, represents the signal transmitted by the acoustic source, represents the source position and represents the nabla differential operator. Applying a Fourier transform to both sides of Eq.(2.1) one can obtain the so called Helmholtz equation:
 (2.2)

where

and

Let's consider a plane wave-like approximation to the solution of Eq.(2.2) and write that [4]
 (2.3)

where represents a slowly changing wave amplitude, and stands for a rapidly evolving phase; the surfaces with constant represent the wavefronts; analogously, the surfaces with constant are called timefronts. By placing Eq.(2.3) into the homogeneous form of Eq.(2.2), considering the high frequency approximation

(where ) and separating the real and imaginary terms of the equation, one can obtain the Eikonal equation:
 (2.4)

and the transport equation:
 (2.5)

The following sections will describe the solution of Eqs.(2.4)-(2.5).

Orlando Camargo Rodríguez 2012-06-21