  The first author proved that the harmonic convolution of a normalized right
half-plane mapping with either another normalized right half-plane mapping or a
normalized vertical strip mapping is convex in the direction of the real axis.
provided that it is locally univalent. In this paper, we prove that in general
the assumption of local univalency cannot be omitted. However, we are able to
show that in some cases these harmonic convolutions are locally univalent.
Using this we obtain interesting examples of univalent harmonic maps one of
which is a map onto the plane with two parallel slits.
