He soon also showed they were equivalent to the existing mathematical formalism, a matrix approach created by Werner Heisenberg. Although the foregoing considerations help us to understand wave propagation in piles, they cannot be applied to practical pile driving problems because they do not take into consideration important physical properties of these systems.� Some of these include but are not limited to the following: Existence of dampening, both at the toe, along the shaft, and in all of the physical components of the system.� In theory, inclusion of distributed spring constant and dampening along the shaft could be simulated using the Telegrapher's wave equation, but other factors make this impractical also.

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