Studies of stationary features in jets: BL Lacertae II. Trajectory reversals and superluminal speeds on sub-parsec scales

High-resolution VLBI observations revealed a quasi-stationary component (QSC) in the relativistic jets of many blazars, which represents a standing recollimation shock. The VLBA monitoring of the BL Lacertae jet at 15~GHz shows the QSC at a projected distance of about 0.26~mas from the radio core.} We study the trajectory and kinematics of the QSC in BL Lacertae on sub-parsec scales using 15~GHz VLBA data of 164 observations over 20 years from the MOJAVE program and 2~cm VLBA Survey. To reconstruct the QSC's intrinsic trajectory, we use moving average and trajectory refinement procedures to smooth out the effects of core displacement and account for QSC positioning errors. We identified 22 QSC reversal patterns with a frequency of $\sim 1.5$ per year. Most reversals have an acute angle $<90\degr$ and few have a loop-shaped or arc-shaped trajectory. Where observed, combinations of reversals show reversible and quasi-oscillatory motion. We propose a model in which a relativistic transverse wave passes through the QSC, generating a short-lived reverse motion, similar to the transverse motion of a seagull on a wave. According to the model, relativistic waves are generated upstream and the reverse motion of the QSC is governed by the amplitude, velocity and tilt of the wave as it passes through. The apparent superluminal speeds of the QSC ($\sim 2\,c$) are then due to the relativistic speed of the jet's transverse wave ($<0.3\,c$ in the host galaxy rest frame) combined with the relativistic motion towards the observer. The measured superluminal speeds of QSC indirectly indicate the presence of relativistic transverse waves, and the size of the QSC scattering on the sky is proportional to the maximum amplitude of the wave. (abbreviated)