In this paper, we present experimental results and reveal that strong perturbations are not necessary for elastic instability to occur in straight-channel, inertialess, viscoelastic flows at high elasticity. We show that a non-normal-mode bifurcation is followed by chaotic fluctuations, self-organized as streamwise streaks, and elastic waves due to weak disturbances generated by a small cavity at the center of the top channel wall. The chaotic flow persists in the transition, elastic turbulence, and drag reduction regimes, in agreement with previous observations for the case of strong perturbations at the inlet. Furthermore, the observed elastic waves propagate in the spanwise direction, which allows us to confirm the elastic waves' linear dispersion relation directly. In addition, the spanwise propagating elastic wave's velocity depends on Weissenberg number with the same scaling that was previously observed for streamwise propagating waves, although their velocity magnitude is significantly smaller than what was previously observed for the streamwise ones.