The Direct Solution Method (DSM) (Geller et al. 1990; Geller & Ohminat
o 1994) is a Galerkin weak-form method for solving the elastic equatio
n of motion. In previous applications of the DSM to both laterally hom
ogeneous and laterally heterogeneous media the vertically dependent pa
rt of the trial functions has been either linear splines (e.g. Cummins
et al. 1994a, b) or the vertically dependent part of modal eigenfunct
ions (e.g. Hara, Tsuboi & Geller 1993). In this paper we formulate the
DSM using analytic trial functions which are solutions of the homogen
eous (source-free) equation of motion in locally homogeneous portions
of the medium. We present explicit formulations for SH and P-SV wave p
ropagation in an isotropic, laterally homogeneous, plane-layered model
. The trial functions so that it is easy to satisfy continuity of disp
lacement at internal interfaces. For the laterally homogeneous SH prob
lem we first find a set of R + 1 analytic trial functions that satisfy
continuity of displacement at the R - 1 internal interfaces between t
he R layers. Each of the trial functions is non-zero in at most two la
yers. We then solve the weak form of the equation of motion, which in
effect enforces the upper and lower boundary conditions, and continuit
y of traction at the internal interfaces. The trial functions are chos
en so that the equation of motion becomes a tridiagonal (R + 1) x (R 1) system of linear equations. For the P-SV problem we define a set o
f 2R + 2 analytic trial functions that satisfy continuity of displacem
ent, but not continuity of traction, at internal interfaces; the trial
functions are chosen so that the equation of motion then becomes a (2
R + 2) x (2R + 2) system with a bandwidth of 7. In contrast, previous
global solution methods (e.g. Chin, Hedstrom & Thigpen 1984; Schmidt &
Tango 1986), which solve simultaneously for both internal continuity
of displacement and traction as well as the external boundary conditio
ns, solve a 2R x 2R system of linear equations for the SH problem or a
4R x 4R system for the P-SV problem, each having approximately twice
the bandwidth of our systems of equations. We show that through an app
ropriate choice of the form of the homogeneous solutions in each layer
our approach can also be readily incorporated into strong-form global
solution methods, thereby leading to exactly the same system of equat
ions obtained by our weak form derivation. We also present the DSM equ
ation of motion for a plane-layered medium composed of a combination o
f fluid and solid layers. The dependent variable in the fluid layers i
s a scalar quantity proportional to the pressure change, while the dep
endent variable in the solid is the displacement. Continuity of displa
cement and traction at fluid-solid boundaries is enforced by augmentin
g the weak-form operator by appropriate surface integrals.