The uniqueness theorem of Simon & Ursell (1984), concerning the linearized
two-dimensional water-wave problem in a fluid of infinite depth, is extende
d in two directions. First, we consider a two-dimensional geometry involvin
g two submerged symmetric bodies placed sufficiently far apart that they ar
e not confined in the vertical right angle having its vertex on the free su
rface as the theorem of Simon & Ursell requires. A condition is obtained gu
aranteeing the uniqueness outside a finite number of bounded frequency inte
rvals. Secondly, the method of Simon & Ursell is generalized to prove uniqu
eness in the axisymmetric problem for bodies violating John's condition pro
vided the free surface is a connected plane region.