  Long linear wave transformation in the basin of varying depth is studied for
a case of a convex bottom profile in the framework of one-dimensional shallow
water equation. The existence of travelling wave solutions in this geometry and
the uniqueness of this wave class is established through construction of a 1:1
transformation of the general 1D wave equation to the analogous wave equation
with constant coefficients. The general solution of the Cauchy problem consists
of two travelling waves propagating in opposite directions. It is found that
generally a zone of a weak current is formed between these two waves. Waves are
reflected from the coastline so that their profile is inverted with respect to
the calm water surface. Long wave runup on a beach with this profile is studied
for sine pulse, KdV soliton and N-wave. Shown is that in certain cases the
runup height along the convex profile is considerably larger than for beaches
with a linear slope. The analysis of wave reflection from the bottom containing
a shallow coastal area of constant depth and a section with the convex profile
shows that a transmitted wave always has a sign-variable shape.
