The initial pressure wave measured at modest distances from an underwater explosion is often modeled as a spherical shock wave with an exponential decay. A closed-form analytical ’’weak-shock’’ solution for the subsequent propagation of such a wave has been obtained. The resulting simple formulas for peak pressure and decay constant as function of reduced range allow the prediction of the amplitude and initial slope of the wave given only the amplitude and decay constant of the original exponential shock and the density, sound speed, and parameter of nonlinearity of the water. The results are in good agreement with the Kirkwood–Bethe theory, existing measurements, and the widely used experimentally based semi-empirical similarity formulas. An expression which gives a close approximation to the shape of the waveform as a function of distance is also derived.

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