fv\'en
waves propagating along the longitudinal component of the field.

In a plasma dominated by the magnetic field, Alfv\'en waves are
transverse displacements of the field (and, perforce, of the plasma),
analogous to waves on a whip. The tension is provided by the magnetic
field $\rm (\propto B^2)$, and the wave velocity is proportional to
the square root of the tension divided by the (relativistic) mass
density. Alfv\'en waves have been employed in various astronomical
contexts, including the acceleration of cosmic rays \citep{Fer49},
the solar wind \citep{BDS69}, the Jupiter-Io system \citep{GL69},
turbulence in the ISM \citep{GS97}, the bow shock of Mars \citep{ELC10},
and the solar atmosphere\citep{Mci11}. In our case they are transverse
waves on a relativistically-moving beam of plasma threaded with a helical
magnetic field.  The appropriate formulas for the phase speeds of the
MHD waves are given in the Appendix of Paper I.

Changes in the ridge lines of BL Lacs are also seen frequently.
\citet{Bri10a} showed that in 1.4 years the BL Lac object 0735+178
changed from having a ``staircase'' structure to being straight, and that
there were prominent transverse motions.  \citet{Bri10b} also studied
1803+784 and described various models that might explain the structure.
\citet{Per12} studied the ridge line in 0836+710 at several frequencies
and over a range of epochs.  They showed that the ridge line corresponds
to the maximum pressure in the jet.  They discussed the concept of
transverse velocity, and concluded that their measured transverse motions
are likely to be caused by a ``moving wave pattern''; this was elaborated
in \citet{Per13}.  In our work here on BL Lac we also see transverse
motions, but their patterns move longitudinally and we identify them as
Alfv\'en waves.  We calculate the resulting transverse velocity of the
wave motion and show that it is non-relativistic.

It has been more customary to discuss the fast radio components in
a relativistic jet in hydrodynamic (HD) terms. We note here only a
few examples of this.  The shock-in-jet model \citep{MG85, Mar14}
was used by Hughes, Aller, \& Aller (1989a, 1989b, 1991) to develop
models of several sources, including BL Lac \citep{HAA89b} and 3C\,279
\citep{HAA91}. \citet{LZ01} recognized two threads of emission in
3C\,273 that they explained with Kelvin-Helmholtz instabilities, and
this was developed more by \citet{PLM06}.  \citet{HWG05} discussed the
patterns and motions in 3C\,120 in terms of helical instability modes.
In all these studies the magnetic field is needed of course for the
synchrotron radiation, but it also is explicitly used to explain observed
polarization changes as due to compression of the transverse components
of magnetic field by the HD shock. But in these works the magnetic
field has no dynamical role in the jet. On the contrary, in this paper,
as in Paper I, we assume that the dynamics in the jet are dominated by
the magnetic field.

\begin{figure*}[t]
\includegraphics[scale=0.9,angle=270,trim=1.8cm 0cm 5cm 0cm]{ridge_all_octo.ps}
\caption{Ridge lines for BL Lac, 1995.26 - 2012.94. Successive
panels are adjacent in time.  Epochs are identified by color. In each panel
the first occurence of a color is further identified as the solid
line, the next occurence as a dashed line, and the third occurence,
when it exists, as a dotted line. The core is shown as the solid dot,
and the semi-circle is drawn 0.25 mas from the core. In all cases the
RCS is close to the circle.
\label{fig:ridge_all}}
\end{figure*}

The plan for this paper is as follows. In Section~\ref{sec:observations}
we briefly describe the observations. The definition of the ridge line of
a jet is considered in Section~\ref{sec:ridgelines}, and the transverse
waves and their velocities, including the behavioral change in 2010,
are presented and discussed in Section~\ref{sec:waves}. Excitation
of the waves by changes in the P.A. of the RCS is considered in
Section~\ref{sec:excitation}.  In Section~\ref{Xverse_mhd} we identify
the waves as Alfv\'en waves, discuss their properties, and present 
simple models of the system.

For BL Lac $z=0.0686$, and the linear scale is $\rm 1.29~pc~mas^{-1}$.  An
apparent speed of $\rm 1~mas~yr^{-1}$ corresponds to $\rm\beta_{app}=4.20$.

\section{Observations}  
\label{sec:observations}

For this study of BL Lac we use 114 epochs of high-resolution
observations made with the VLBA at 15 GHz, between 1995.27 and
2012.98.  Most of the observations (75/114) were made under the MOJAVE
program\footnote{\url{http://www.physics.purdue.edu/astro/MOJAVE/}}
\citep{LH05}, a few were taken from our earlier 2-cm program on the VLBA
\citep{KVZ98}, and the rest were taken from the VLBA archive.

The data were all reduced by the MOJAVE team, using standard
calibration programs \citep{Lis09}. Following the reduction to fringe
visibilities we calculated three main products at nearly every epoch: 
\begin{description} 
\item[(1)]  An image, consisting of a large number of
``clean delta functions'' produced by the algorithm used for
deconvolution, convolved with a ``median restoring beam'', defined in
Section~\ref{sec:ridgelines}.
\item[(2)]  A model, consisting of a set of Gaussian
``components'' found by model-fitting in the visibility plane; each
component has a centroid, an ellipticity, a size (FWHM), and a flux
density. The Gaussians are circular when possible.  The total set of
components sums to the image, but in this paper we only use components
that have been reliably measured at four or more epochs, have flux density
$\rm >20~mJy$, and can be tracked unambiguously from epoch to epoch.
A typical epoch shows 4-6 of these ``robust'' components.  The RCS is
a permanent component and, together with the core, usually produces
more than half of the total flux density from the jet.  The centroids
of the robust components for each epoch are plotted on the images in
Figure~\ref{fig1}.

The centroid locations are measured relative to the core, which we take
to be the bright spot at the north end of the source; it usually is
regarded as the optically-thick ($\tau = 1$) region of the jet. In
principle, the core can move on the sky. We considered this in Paper I,
and concluded that any motions are less than $\rm 10~\mu as$ in a few
years, and they were ignored. Our positional accuracy is conservatively
estimated as $\rm \pm 0.1~mas$, and in this paper we again ignore any
possible core motions.
\item[(3)] The ridge line, shown in Figure 1 and discussed
in Section~\ref{sec:ridgelines}.  
\end{description} 
The image, the components, and the ridge line are not independent, but
each is advantageous when discussing different aspects of the source. In
most cases the ridge line runs down the smallest gradient from the peak
of the image, and the centroids of the components lie on the ridge
line. However, when the jet has a sharp bend the algorithm can fail, as
in Figure~\ref{fig1}c. This is discussed in Section~\ref{sec:ridgelines}.

The components move in a roughly radial direction, and plots of
$r(t)$ as well as the sky (RA--Dec) tracks are shown in Paper I and in
\citet{Lis13}.  The tracks cluster around an axis at $\mathrm{P.A.} = -166\ifmmode^\circ\else$^\circ$\fi$
and appear to emanate from a strong quasi-stationary component, C7, that
we identified as a recollimation shock (RCS) in Paper I.  The moving
components have superluminal speeds; the fastest has $\rm \beta_{app}
= 10\pm 1$ in units of the speed of light.  \citep{Lis13}


\section{The Ridge Lines}  