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Let us consider the equation (32) for *n* = 0:

With accordance to equation (23), this is a linear algebraic
equation with degenerate matrices. It may be solved only in
the case when is an eigenvector.
We restrict ourselves with the situation when the eiconal
equation for P-waves is valid. In this case, the eigenvector is
so vector can be
expressed in the form . In
order to specify the amplitude *A*_{0} let us consider the
equation (32) for *n* = 1:
| |
(34) |

But any vector can be decomposed into the sum
where . Both vectors and are eigenvectors connected with eigenvalues
and correspondently, therefore,

We see that the left side of the equation (34) is
orthogonal to the vector . It means that
equation (33) can be resolved if and only if

| |
(35) |

Inserting expression in equation (33) into the last equation, we use the
property of the operator , if *s* is the
natural parameter of a ray , then for
any scalar function

where means differentiation along the ray that
intersects the given point. After simple manipulation we
derive from equation (35) the transient equation for the amplitude
*A* = *A*_{0}/*v*:
| |
(36) |

which have solution
The well-known notion of geometrical spreading is connected with by relation
(S.V. Goldin, 1986)

consequently,
| |
(37) |

Analogous formula is true for S-waves.
The equation (37) describes how the amplitude of the senior
discontinuity of a wave is changing along a ray path.

If the value of *J* is positive in all points of a given ray , then along the ray

| |
(38) |

where and is the tangent-vector of the ray. And
if *J* < 0, then
The situation when *J* = 0 means that at *q*'<*q*_{0}, .

** Next:** Fundamental solutions
** Up:** 4: CONNECTION WITH RAY
** Previous:** Ray method
Stanford Exploration Project

1/13/1998