Mechanism of sound emission produced by enclosed cavity upon a sphere entering the water

Abstract

A mathematical model for the acoustic pressure field induced by an enclosed cavity upon a solid sphere impacting into water is established. The relation between sound emission, cavity ripples, and the falling sphere is elaborated analytically. To describe the water domain above the sphere, physical and co-moving cylindrical coordinate systems are introduced. The water domain is analyzed by the perturbation method and potential flow theory. The air inside the cavity is assumed to be uniform and adiabatic. In the physical coordinates, the zeroth-order cavity surface is governed by the Rayleigh–Plesset Equation and is cylinder-like. In the co-moving coordinates, the sphere is stationary and subjected to an upward free stream. The slender cavity geometry permits linear surface waves with a vertical normal-mode solution, which corresponds to the first-order cavity ripples. The formula of ripples suggests that the cavity nose is a falling, pulsating, flat, cylindrical acoustic source in water. The induced acoustic wave is formulated in another spherical coordinate system. The sound frequency equals the vibration frequency of the cavity nose and also equals the formation/propagation frequency of the ripples. Moreover, the natural frequency of the cavity nose and the natural normal mode of the ripples are formulated. A water entry experiment is implemented and validates the theory. The normal mode of the cavity ripples, the criteria of the sound emission, and the initiation of the ripples and sound emission are discussed.