  We analyze plane waves in a model of quantum mechanics in a three dimensional
noncommutative (NC) space $R^3_{\lambda}$. Signature features of NC models are
impossibility of probing distances smaller than a certain length scale
{\lambda} and a presence of natural energetic cut-off at energy scale of order
$1/{\lambda}^2$ (in convenient units). We analyze consequences of such
restrictions on a 1 dimensional Casimir effect. The result shows resemblance to
flux tube potential for quark-antiquark pairs and to effective bosonic string
theories with L\"uscher terms. Such behavior might effect the radius of
possible compact (fuzzy) dimensions.
