The general procedure of the perturbation method is to identify a small parameter (i.e., ), such that when this parameter is set to zero the problem (for example, the differential equation) becomes soluble. Thematically, the approach decomposes a tough problem into an infinite number of relatively easy ones. Hence, perturbation method is most useful when the few first steps reveal the important features of the solution and the remaining ones give small corrections. Another feature of the perturbation method is that it does not add additional solutions to the solutions already obtained from the unperturbed medium (i.e., background medium). The method simply perturbes those solutions obtained for the background medium (in our case isotropic) to accommodate the perturbation in the medium.

From equation (7), one parameter that can clearly be small is . Setting =0 yields the acoustic wave equation for elliptically anisotropic media. For isotropic (or elliptically anisotropic) acoustic media, we have two complex-conjugate solutions; one corresponds to outgoing waves and the other to incoming waves. Perturbation from a background medium of =0 will result in only two solutions as well.

According to the perturbation theory (Bender and Orszag, 1978), the solution of equation (7) can be represented in a power-series expansion in terms of . Since in practical applications, this power series is truncated, we use only the first four terms. Therefore, to find the perturbed solution, we write

as a trial solution of equation (7) for the two-dimensional problem. In addition, we consider as a trial solution of equation (11). Because is a variable, we can set the coefficients of each power of separately to equal zero. This will provide a recursive method of solving for the coefficients,Substituting the trial solution into the partial differential equation (11), and considering only the term of zero-power in yields

(32) |

The coefficient of first power in yields

(33) |

(34) |

(35) |

(36) |

C

10/9/1997