Pressure Field in the Liquid around a Vertically Heaving Sphere

Abstract

The calculation of the pressure field around a periodic vertically heaving sphere half-submerged in fluid of infinite depth is described. To establish the formula for the fluctuating pressure, the forces—which are caused by the heaving body and which depend on the added mass and damping coefficients—must be known. The general expression for the fluctuating force is derived from the velocity potential. Franz converted monopole and dipole expansions to a useful form by introducing certain weighting functions. He further developed an expression for the fluctuating pressure and calculated the resulting pressure directly below the sphere. As an extension to the Franz presentation, the pressure is calculated for different angles from the vertical-heave direction around the submerged hemisphere. The distance from the sphere is taken in the geometrical farfield, but much less than the acoustic wavelength. The plotted weighting functions, pressure components, and average pressure for different angles indicate that they are valid only for restricted-heave frequency regions, which are smaller for greater angles. This is most evident for points on the surface, i.e., for angles near 90°. In the limits of high or low frequency, the classical results are obtained.