New notations and definitions

Until now the proof was pure physics. But because in seismology the depth axis is ``special,'' we will change notations. Axis x₃ will be denoted by z, and the 73#73 will denote the surface position vector with the components (x₁, x₂). Also, the Laplacian will be the operator,  

 464#464 (192)

We will use the Fourier transform of the pressure field along the time axis:  

 465#465 (193)

The following property of the Fourier transform will be needed:  

 466#466 (194)

The spatial frequency is defined as:  

 467#467 (195)

Let 468#468 be a spatial average of v in the medium, a known constant that does not depend on 73#73 or z. We also define  

 469#469 (196)

and the function  

 470#470 (197)

The index notation for derivatives will be used from now on. The symbol 471#471 will denote the phrase ``for all.''