[[spi:math]] (26) At the ionosphere z=[[spi:math]] we have the boundary condition (20), while at the equator we have the boundary condition (18). It must be noted that (18), with the subsidiary condition (19), assumes the beam spread in the z direction (along magnetic field lines) to be small so that the Alfven wave is assumed to be generated by a sheet source at the equator moving with the entire momentum of the beam. A cursory look at Figure 1 will indicate that this assumption is at best an abstraction since the beam plasma will spread along the magnetic field line at about one tenth the beam speed. Thus, the Alfven wave source is distributed rather than sheet-like, albeit the parallel spreading speed is much smaller than the Alfven speed. Since our causal solution is in effect a Green’s function solution, the construction of a solution for distributed sources is straightforward but somewhat tedious. In FY80, we have made stu- dies of the distributed Alfven wave source model, but the numerical analysis is so complex that it would be more appropriate to confine ourselves to the results of the sheet-source model in this report. Hopefully, we will be able to report on the distributed source analysis as we proceed with further analyses of SPS magnetospheric effects. Without going into the details of how the causal Laplace transform of (23) can be inverted exactly so as to obtain the solution [[spi:math]]. in terms of 24 of the jth segment. At the jthinterface zj between segments, we have the boundary conditions [[spi:math]] (27)
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