Without A Trace

Joseph Malkevitch
Department of Mathematics
York College (CUNY)
Jamaica, New York 11451

Background Information:

We are accustomed to the fact that many artifacts of older civilizations are deeply buried under forest or soil at the current time. Imagine that humans abandon a certain urban or suburban community. (Perhaps the populations migrated to some other locale.) How long would it take before the signs of life, houses, streets, etc. were no longer visible?

Problem:

Here is a crude attempt to model this situation:

Suppose one has a very "small" rectangular shaped "forest" where fully grown identical trees of a particular species that were planted at exactly the same time are growing. This forest is enclosed in a walled somewhat larger rectangular box open at the top, with the forest symmetrically placed in the center of this walled box. Rain can come in from the top, but the ground absorbs all the water.

Despite this artificial environment the forest grows leaves every season, but these leaves accumulate at the bottom of the "box." The floor of the box is large enough so that the ecosystem there continues to operate, and some of the leaves decay and compactify. At the end of each season the winds distribute the new leaves uniformly throughout the floor of the larger box.

How long will it take for the leaves to reach height h in the larger box?

Reference:

You may wish to take a look at "The Leaf Problem" which constitutes a "warm-up" for this problem.