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The effects of both density stratification and magnetic field expansion ontorsional Alfven waves in solar spicules are studied. Also, theireigenfrequencies, eigenfunctions, and the period ratio P_1/P_2 are obtainedwith a novel mathematical method. We showed that under some circumstances thisratio can approach its observational value even though it departs from itscanonical value of $2$. Moreover, Eigenfunction height variations show that theoscillations amplitude are increasing towards higher heights which is inagreement with the observations. This means that with a little increase inheight, amplitude of oscillations expands due to the significant decrease inthe density.

Torsional Alfven waves and the period ratio P_1/P_2 in spicules