Summary: 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 for finding ordinary differential equations (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 \(\gamma^{damp} = 1\), which is the same as the dry wall result.