  We study the kinematics of ridge lines on the pc-scale jet of the active
galactic nucleus BL Lac. We show that the ridge lines display transverse
patterns that move superluminally downstream, and that the moving patterns are
analogous to waves on a whip. Their apparent speeds $\beta_\mathrm{app}$ (units
of $c$) range from 3.9 to 13.5, corresponding to
$\beta_\mathrm{wave}^\mathrm{gal}= 0.981 - 0.998$ in the galaxy frame. We show
that the magnetic field in the jet is well-ordered with a strong transverse
component, and assume that it is helical and that the transverse patterns are
Alfv\'en waves propagating downstream on the longitudinal component of the
magnetic field. The wave-induced transverse speed of the jet is
non-relativistic ($\beta_\mathrm{tr}^\mathrm{gal} \lesssim 0.09$). In 2010 the
wave activity subsided and the jet then displayed a mild wiggle that had a
complex oscillatory behaviour. The Alfv\'en waves appear to be excited by
changes in the position angle of the recollimation shock, in analogy to
exciting a wave on a whip by shaking the handle. A simple model of the system
with plasma sound speed $\beta_\mathrm{s}=0.3$ and apparent speed of a slow MHD
wave $\beta_\mathrm{app,S}=4$ yields Lorentz factor of the beam
$\Gamma_\mathrm{beam} \sim 4.5$, pitch angle of the helix (in the beam frame)
$\alpha\sim 67^\circ$, Alfv\'en speed $\beta_\mathrm{A}\sim 0.64$, and
magnetosonic Mach number $M_\mathrm{ms}\sim 4.7$. This describes a plasma in
which the magnetic field is dominant and in a rather tight helix, and Alfv\'en
waves are responsible for the moving transverse patterns.
