Implementation

>From this point the development follows the lines given in the SAFARI Users Guide[4] and subsequent updates. Scalar and vector potentials are introduced so that

In terms of these potentials, and in the absence of sources, the equations of motion become

These equations are first subjected to Fourier transform in time and either Fourier transform (for plane strain) or Hankel transform (for axial symmetry) in the plane perpendicular to the depth coordinate z. We show details assuming plane strain with motion restricted to the xz-plane (the axial symmetry case is similar and follows the outline in the SAFARI Users Guide). In this event the vector potentials have only a single nontrivial component, , so the displacement components take the form

Furthermore, the constitutive equations, in terms of the potentials, become

The solution of the transformed equations take the form

where and are the transformed potentials and , , and are functions of angular frequency and wave number s. Also

and

where h_f, h_s, and k are respectively the complex wavenumbers associated with fast, slow and shear waves in the Biot medium at angular frequency . For the shear wave we have

while the wave numbers associated with the fast and slow waves are given by

where

and the two compression wave numbers are ordered so that .

In terms of the six degrees of freedom , , and for a porous sediment layer, the transformed displacement and stress components are

Boundary conditions to be satisfied at each interface involve continuity of pore fluid normal displacement and pressure, and continuity of displacement and traction components for the frame. To complete table 1 of the SAFARI Users guide we add the following table entries where the symbol means the parameter is not involved, 0 means it should be set to zero, and = means continuity is enforced at the interface. In the following table the first four columns correspond to the same entries in the Users Guide table, while the last column corresponds to the negative of the traction component in adjacent fluid layers.

Type	Field parameter
	w - wf	u			wf	p
Biot/vacuum			0	0		0
Biot/fluid	=		=	0		=
Biot/solid	=	=	=	=	0
Biot/Biot	=	=	=	=	=	=