The Growth of Disturbances in Unstable Infiltration Flows

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Abstract
The stability analysis of the delta‐function model of infiltration yields a maximum wave‐number for unstable disturbances, Mcrit, and a wave‐number for maximum amplification, M*. The “column effect,” i.e. the suppression of instability in laboratory columns, depends on Mcrit; and the dimensions of fingers in unstable flows can be expected to depend on M*. When instability is induced by a sudden jump of G, the water pressure gradient behind the wetting front, Mcrit and M* may be evaluated simply and directly. When, on the other hand, instability emerges during gradual increase of G, it is necessary to study the amplification of disturbances during the whole course of increase of G through positive values.Such a study is made for one example of instability induced by air compression and for one where it is induced by increase of conductivity with depth. Both exhibit qualitatively similar patterns of amplification: disturbances of small M (wave‐number) begin to be amplified sooner, but are ultimately overtaken by some disturbances of larger M starting later. Definite values of M* are found, and the results discussed with special reference to fingering and to the column effect.

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