Abstract: In this paper we develop a variational theory to study the dynamic
properties of ultracold Bose gas in a funnel external potential. We obtain one-dimensional nonlinear equation which describes the dynamics of transverse tight confined bosonic gas from three-dimension to one-dimension, and find one-dimensional s-wave scattering length which depends on the shape of
transverse confining potential. If the funnel trapping potential is strong enough at zero temperature, all transverse excitations are frozen. We find the dynamic equation which describes the Tonks-Girardeau gas and present a qualitative analysis of the experimental accessibility of the Tonks-Girardeau gas with funnel-trapped alkalic atoms.

Key words: funnel potential, variational method, ultracold Bose gas, Tonks-Girardeau region