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AVERAGE SURFACE DISPLACEMENT

The average surface displacement caused by the passage of a simple wave, where, for simplification, the wave is seen to impinge on every other surface and undergo interchange (a 0.5 probability), would be twice the distance between surfaces, or 2h.

If one views the passage of a wave as being a bit more complex, such as sometimes impinging on the first surface or skipping any number of surfaces before impinging, where the probabilites involved are as follows:

      INTERCHANGE WITH     ODDS REACHING IT    DISTANCE TRAVELED    PROBABILITY TIMES DISTANCE
        FIRST SURFACE                    0.5                                          1 h                                              0.5 h
        SECOND SURFACE               0.25                                        2 h                                              0.5 h
        THIRD SURFACE                   0.125                                      3 h                                              0.375 h
         FOURTH SURFACE              0.0625                                    4 h                                              0.25 h
         FIFTH SURFACE                   0.03125                                  5 h                                              0.15625 h
         SIXTH SURFACE                   0.015625                                6 h                                              0.09375 h
         SEVENTH SURFACE             0.0078125                              7 h                                              0.0546875 h
         EIGHTH SURFACE                0.0039063                              8 h                                              0.03125 h
         NINTH SURFACE                   0.0019531                              9 h                                              0.0175781 h
         TENTH SURFACE                  0.0009766                            10 h                                              0.0097656 h

TOTAL PROBABILITY TIMES DISTANCE = 1.99 h

In examining a set of ten surface combinations, a wave will most likely travel 1.99 h, thus leaving its surface displaced by the same amount.  Given a set of an infinite number of surfaces (all possibilities), a surface would be displaced by 2.0 h, though pragmatically impossible.

What this means is, in the application of this very simple mathematical analysis, one can expect a wave to travel through surfaces, leaving them displaced on average 2 h, even though the distance between is 1 h.

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