Series expansion methods are used to study directed bond percolation clusters on the square lattice whose lateral growth is restricted by a wall parallel to the growth direction. The percolation threshold is found to be the same as that for the bulk. However, the values of the critical exponents for the percolation probability and mean cluster size are quite different from those for the bulk and are estimated by and respectively. On the other hand the exponent characterizing the scale of the cluster size distribution is found to be unchanged by the presence of the wall.

The parallel connectedness length, which is the scale for the cluster length distribution, has an exponent which we estimate to be and is also unchanged. The exponent of the mean cluster length is related to and by the scaling relation and using the above estimates yields to within the accuracy of our results. We conjecture that this value of is exact and further support for the conjecture is provided by the direct series expansion estimate .