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We analyze plane waves in a model of quantum mechanics in a three dimensionalnoncommutative (NC) space $R^3_{\lambda}$. Signature features of NC models areimpossibility 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 suchrestrictions on a 1 dimensional Casimir effect. The result shows resemblance toflux tube potential for quark-antiquark pairs and to effective bosonic stringtheories with L\"uscher terms. Such behavior might effect the radius ofpossible compact (fuzzy) dimensions.

NC plane waves, Casimir effect and flux tube potential with L\"uscher terms