  Solutions for a class of wave equations with effective potentials are
obtained by a method of a Laplace-transform. Quasinormal modes appear naturally
in the solutions only in a spatially truncated form; their coefficients are
uniquely determined by the initial data and are constant only in some region of
spacetime -- in contrast to normal modes. This solves the problem of divergence
of the usual expansion into spatially unbounded quasinormal modes and a
contradiction with the causal propagation of signals. It also partially answers
the question about the region of validity of the expansion. Results of
numerical simulations are presented. They fully support the theoretical
predictions.
