Fig. 2: Simulation results and mechanism of active traveling wave.

a Simulated time sequence of two traveling undulating bacterial streams. Black arrows indicate traveling directions. b Simulated space-time graph of active traveling wave along the +x-axis. c Corresponding simulated power spectrum of the traveling wave. ω and k refer to normalized frequency and wavenumber, respectively. d Mechanism of active traveling wave. The solid black line is the bacterial wave in the splay region. Red and blue dashed lines represent rough direction of director and bacteria, respectively. Curved arrows represent different torques at two regions ① and ②, which cause different viscosities and thus a rectified net driving force for the wave propagating. The dashed black line is the traveling bacterial wave at the next moment. e Mechanism of standing bacterial wave in uniform director field compared to that in (d). f Spatial distribution of the angle \(\beta\) (color bar) between bacteria and LC director in splay director field. The solid black line is the center line of the bacterial wave in the splay region. Brown arrows represent the orientation of bacteria with angle \(\theta\). Black short bars are the director field with angle \(\psi\). g Spatial distribution of the angle \(\beta\) (color bar) in a uniform director field. h Spatial distribution of the torque in the splay region. Brown and black arrows represent the bacteria's orientation and the force induced by the torque, respectively. i Spatial distribution of the torque and force in a uniform director field.