A new ansatz is formulated, whereby a large class of focus wave mode (FWM)-type finite energy localized wave (LW) solutions to the axisymmetric 3-D Klein-Gordon equation is obtained by means of a dimension (or coordinate)-reduction technique. Each of these solutions consists of a product of a specific solution to the 1-D scalar wave equation, with coordinates z and t, and, essentially, an arbitrary analytic solution to the 1-D Klein-Gordon equation, with coordinates Z and T appropriately defined in terms of z, t and the polar radial coordinate p. In the absence of dispersion, the same formalism, but with a different definition of the coordinates Z and T, can be used to obtain FWM-type finite energy localized wave solutions to the 3-D scalar wave equation. In this case, the aforementioned ansatz is intimately related to a technique due to Bateman [1, 2].