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In a dispersive medium, the appearance of the steady-state part of the signal is preceded by oscillations known as precursors. These early oscillations are the product of the interrelated effects of phase dispersion and frequency dependent attenuation. Inside water, the attenuation rate of the Brillouin precursor is sub-exponential, following the inverse square-root of the distance traveled. Based on that, a near-optimal pulse that could achieve this attenuation rate, and, hence, would lend itself to underwater detection and communication applications, was recently proposed. The ldquooptimalityrdquo of this pulse is shown to be related to the temporal support of the pulse and its spectral characteristics, rather than its shape. A family of alternative pulses is found to have the low attenuation feature of the ldquooptimalrdquo pulse, as they eventually evolve into the Brillouin precursor itself shortly after they enter water. In addition, this work considers the practical case when such a pulse would be generated in air, would impinge onto an air-water interface and then propagate inside water. It is shown how the presence of the interface affects the attenuation rate of the pulse inside water and a simple way to recover its low attenuation rate is suggested. The finite-difference time-domain technique is employed in all the simulations.