Wave pattern in the wake of an arbitrary moving surface pressure disturbance

Abstract

We study the problem of wave pattern in the wake of an arbitrary surface pressure disturbance that moves forward at constant speed U in deep water. We seek the dependence of the location of the maximum amplitude of waves upon the pressure distribution and the Froude number F ≡ U/√gL , where L is the characteristic length of the pressure disturbance and g is the gravitational acceleration. We show by theoretical analysis and direct numerical evaluation that half of the included angle (φ{symbol}max) of the V-shape corresponding to the maximum amplitude of the waves in the wake at large Froude numbers behaves asymptotically as φmax = CF -a for F > Fc, with the constant a, coefficient C, and threshold value of Froude number Fc all being functions of the pressure distribution. It is found that for most pressure disturbances, a equals 1, but a can equal 2 for special non-smooth pressure disturbances. The condition in terms of the order of discontinuity and distribution shape of the pressure disturbance for the result of a = 2 is provided. These findings imply that for ship wakes, φ{symbol}max generally decreases with increasing F at large Froude numbers, while the exact value of φ{symbol}max is dependent on ship geometry and F.