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 A0 let us consider the equation (32) for n = 1:
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
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 = A0/v:
The well-known notion of geometrical spreading is connected with by relation (S.V. Goldin, 1986)
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
The situation when J = 0 means that at q'<q0, .