  The effect of the Doppler shift is studied in a model for the
$\alpha$-$\beta$ bands of Sr$_2$RuO$_4$ consisting of two hybridized 1D bands.
Assuming a superconducting gap with nodes in the diagonal directions, we
examine the oscillation of the surface density of states and the thermal
conductivity under a rotating magnetic field. Upon varying the strength of the
hybridization, the oscillation in these quantities is found to exhibit 2D to 1D
crossover. In the crossover regime, which corresponds to the actual
Sr$_2$RuO$_4$, the thermal conductivity exhibits a two-fold-symmetry
oscillation, while the four-fold-symmetry component in the oscillation is
barely detectable.
