  We here present the first analytic effective fly-by (EFB) waveforms designed
to accurately capture the burst of gravitational radiation from the closest
approach of highly eccentric compact binaries. The waveforms are constructed by
performing a re-summation procedure on the well-known Fourier series
representation of the two-body problem at leading post-Newtonian order. This
procedure results in two models: one in the time-domain, and one in the Fourier
domain, which makes use of the stationary phase approximation. We discuss the
computational efficiency of these models, and find that the time-domain model
is roughly twice as fast as a numerical quadrupole waveform. We compare the
time-domain model to both numerical, leading post-Newtonian order, quadrupole
waveforms and numerical relativity fly-by waveforms using the match statistic.
While the match is typically $>0.97$ when compared to the quadrupole waveforms,
it is much lower when comparing to the numerical relativity fly-by waveforms,
due to neglecting relativistic effects within the model. We further show how to
use these individual waveforms to detect a repeated burst source.
