  We investigate the behaviour of the mean size of directed compact percolation
clusters near a damp wall in the low-density region, where sites in the bulk
are wet (occupied) with probability $p$ while sites on the wall are wet with
probability $p_w$. Methods used to find the exact solution for the dry case
($p_w=0$) and the wet case ($p_w=1$) turn out to be inadequate for the damp
case. Instead we use a series expansion for the $p_w=2p$ case to obtain a
second order inhomogeneous differential equation satisfied by the mean size,
which exhibits a critical exponent $\gamma=2$, in common with the wet wall
result. For the more general case of $p_w=rp$, with $r$ rational, we use a
modular arithmetic method of finding ODEs and obtain a fourth order homogeneous
ODE satisfied by the series. The ODE is expressed exactly in terms of $r$. We
find that in the damp region $0<r<2$ the critical exponent $\gamma^{\rm
damp}=1$, in common with the dry wall result.
