A new solution method for the wave equation in inhomogeneous media

Abstract

A fundamental mathematical algorithm is presented for solving the wave equation in inhomogeneous media. This method completely generalizes the Haskell matrix method, which is the standard method for solving the wave equation in laterally homogeneous media. The Haskell matrix method has been the mathematical basis for many seismic techniques in exploration geophysics. In the method presented the medium is divided into layers and vertically averaged within each layer. The wave equation, within a layer, is then decoupled into an eigenvalue equation of the horizontal coordinate and a wave equation of the vertical coordinate. The eigen‐value equation is solved numerically. The vertical equation is solved analytically, once the eigenvalues are found. The solution throughout the medium is constructed by matching layer solutions at layer interfaces. The solution process of this method is “modular,” in the sense that each layer corresponds to an independent module and all the modules together form the final, total solution. Such a modular solution process has the following advantages. First, in a 2-D problem, for example, each module is a 1-D problem, which is a much simpler problem numerically than the original full 2-D problem. Second, the module solutions can be used repeatedly to form the solution corresponding to different problems. For example, in modeling. only those layers which differ between two models require recalculation. The solution to plane‐wave diffraction by a cylinder is obtained using this method, and it agrees well with the analytical solution.

One-Dimensional Wave Propagation in a Functionally Graded Elastic Medium

“…The exact solution of the problem is given by equation (25) whereas equation (21) gives the asymptotic result. The accuracy of the result would be expected to depend on the number of terms retained in the inner series in equation (21). For the example considered Table 3 shows the comparison of the normalized stress s xx /s o obtained from the exact solution (25) and from equation (21) by retaining a limited number of terms in the inner series.…”

“…There are also purely numerical techniques to treat the problem of wave propagation in inhomogeneous solids. For example, Pai [21] used the method of generalized Haskell matrix to solve the one-dimensional wave equation, whereas Harker and Ogilvy [22] used a ®nite difference technique to obtain the solution of the coherent wave propagation problem for inhomogeneous materials.…”

“…By substituting from equations (28), (29), and Appendix B into equation (26), arranging the expansion to a suitable form for wave summation and inverting the result, one obtains an asymptotic solution similar to equation (21). For example, it may be observed that for large values of p the ®rst term approximations are…”

“…In equation (34) d is de®ned by equation (19). Note that the transforms given by equation (34) are quite similar to equation (20) and would lead to an asymptotic solution in the wave summation form similar to equation (21). Also note that for large values of p one has a ®rst term approximation…”

Comparison of the WKBJ and truncated asymptotic methods for an acoustic medium

“…The limitation in all of these cases is that the solution is given in terms of special functions. In addition I-D synthetic seismograms in a vertically inhomogeneous medium have been calculated using either combinations of transition layers (Berryman, Goupillaud & Waters 1958;Bortfeld 1960;Scholte 1961;Gupta 1965) or layer-matrix formulations (Thomson 1950;Haskell 1953;Wuenschel 1960;Pai 1985;Dietrich & Bouchon 1985).…”