Fig. 3: Mechanism of double traveling waves.

a Schematic of double traveling waves. At first, undulation in the bacterial stream causes wave deformation in LC. As the bacterial wave (green) travels forward, there is a phase lag between the LC wave (red) and the bacterial wave. These two waves attract each other and the LC wave travels as well. Finally, active-passive waves reach a stable traveling state with a steady phase difference. b Different effects exerted by bacteria on LC contributing to the traveling of LC wave. Along the x-axis, the acting effects show different phases of the LC wave. Green and orange solid lines represent bacteria (\({{{\rm{P}}}}_{{{\rm{xx}}}}\)) and LC (\({{{\rm{Q}}}}_{{{\rm{xx}}}}\)) waves, respectively. Dashed lines are contributing effects. FL means flow alignment. CR means corotation with the local vorticity. Adv represents advection. Total means the total effect. c Simulated double traveling waves between crossed polarizers. Red and blue lines represent bacterial waves and LC waves, respectively. Circled numbers with corresponding color mark wave crests and troughs in different waves. Black arrows indicate traveling directions. d Traveling bacterial wave and LC wave without polarizers. Red arrows indicate the bacterial wave and black bars are LC directors. The color bar means the order parameter of Q. The circled numbers and arrows show the traveling wave peaks.