Causality and the radiation condition

✍ Scribed by John V. Wehausen

• Publisher: Springer
• Year: 1992
• Tongue: English
• Weight: 292 KB
• Volume: 26
• Category: Article
• ISSN: 0022-0833
• DOI: 10.1007/bf00043233

✦ Synopsis

For a hydrostatically stable floating body making small oscillations about a fixed position as a result of external forces and moments, it is shown that the radiation condition implies that the motion at time t depends only upon the forces and moments at times ~<t, i.e. that the future does not determine the present.

We wish to consider here one aspect of the motion of a body floating in a heavy fluid. The motions will be assumed to be small enough so that the equations may be linearized. In order to describe the motion of both body and fluid, we shall adopt a right-handed coordinate system with Oz directed against gravity, Ox to the right, and Oy into the paper. The plane Oxy lies in the undisturbed free surface. The small excursions that the body makes about its fixed equilibrium position will be denoted by 19/1, . . . , O~6, where a 1, a2, a 3 represent translational displacements and an, a 5, o~ 6 angular ones. The dynamical constants of the body will be denoted by mik where ml~ = m22 = m33 = m, the mass of the body, and 2 mik = p [r 6i~ -xixk] dV , i,k=4,5,6,