  In this paper we present the power series solution of the Klein-Gordon
equation in the spacetime background of a gravitational wave with amplitude
that decays with distance from the source. The resulting solution describes the
interaction of a scalar plane wave travelling in an arbitrary direction
relative to the direction of propagation of the gravitational wave. This
solution has the unexpected property that as the scalar wave approaches
collinearity with the gravitational wave there is a rapid transition in the
form of the solution. The solution in the collinear limit exhibits a resonance
phenomenon which distinguishes these results from previous analyses involving
plane gravitational wave backgrounds. We discuss in detail the similarities and
differences between the solutions for plane gravitational waves and
gravitational waves with amplitude that decreases with distance from the
source. We give an argument that this solution of the Klein-Gordon equation
only describes the interaction of a gravitational wave with a scalar wave and
that the gravitational wave will not produce a scalar waveform in a vacuum. The
interaction between the gravitational and scalar waves lead to both sinusoidal
time-dependent fluctuations in, and time-independent enhancement of, the scalar
current in the direction of the gravitational wave. Finally, we discuss the
possibility of observable effects of this interaction.
