[[spi:math]], we obtain the exact solution for the velocity field of the Alfven wave at location zi of the ith segment, ai-1 <= zi <=ai where aj is the jthinterface. [[spi:math]] [[spi:math]] and [[spi:math]] 25 [[spi:math]] (28) where [[spi:math]] and [[spi:math]] (29) In (29), [[spi:math]] and where [[spi:math]] can be [[spi:math]] or [[spi:math]] as in indicated in (28). Further Lq(x) in 4 (29) is the Laguerre function of order q and [[spi:math]] is the causal Heaviside function such that [[spi:math]] = 1 if x > 0 and [[spi:math]] otherwise. In (30) [[spi:math]] is the Alfven wave bounce time (30) (31) (32)
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