Step Input Function

A step input function is zero up until time t0, and then increases to a nonzero constant afterwards. That is, it actually looks like a "step":
A Step Function Graph
(Actually, in mathematics we refer to any function that is constant except for one jump like this a step function. However, in this module on lake modelling, we will always mean that the function is zero up until time t0 and increases after that time.)

Therefore, we use a step input function to model a new source of contaminant appearing in the input of a lake. Previously no contaminant was entering the lake, and now contaminant begins to enter at a constant rate. An example would be a manufacturing plant which begins operation and starts dumping waste into a river feeding the lake. In this case the initial time t0 would be when the plant began operation. Before that time, no contaminant entered the lake, and after that time the contaminant entered at a (more or less) constant rate.

In our simulations, we always assume that t0 is time zero and that the contaminant begins being added at the start of the simulation. We enter into the simulators the constant concentration of contaminant in the inflow and then start the simulation.

We might of course use a step input function to model other situations. For example, we might use it to model a sudden increase in contaminant from the inflow, even if the previous input concentration was not negligible, by assuming an initial concentration of contaminant in the lake. We might also use a step model even if the input concentration is not really constant, if we decide that the variations in the input level are negligible over the time frame we are considering. The significant points are that

  1. The input suddenly increases at time t0, and
  2. The input remains essentially constant after time t0.

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John Hoggard <hoggard@math.vt.edu>
Last modified: Fri Jun 27 16:12:26 EDT 1997