Deterministic Response to Free-Surface Waves

Abstract

The relative importance and method of application of the impulse response technique for predicting motions, forces, etc., due to free-surface wave effects on water-borne vehicles, has been discussed by several authors. Controversy exists over the applicability of the law of causality when there is more than one independent variable; that is, the free surface and the body whose response is desired are distributed in space and vary in time. It is shown in this paper that impulse response operators for systems with free-surface wave inputs, in principle, do not satisfy the law of causality. However, by properly positioning the wave input point, the contribution from negative time can be made small. Nomenclature g = gravitational constant h(t) = time history of surface elevation at origin p(x, y, t) = pressure field t = time x, y = orthogonal Cartesian coordinate system f = complex Fourier transform parameter, f = « + iar)(x, t) = free-surface elevation p = fluid density (x} y, 0 = velocity potential E FFECTIVE operation of marine vehicles in the unsteady ocean environment requires analytical methods to predict motions, accelerations, deck wetness, forefoot emergence, etc. Two equivalent approaches have been described for such predictions. St. Denis and Pierson1 presented methods for determining the probabilistic description of the behavior of a vehicle in a random seaway. The complementary procedure presented by Cummins 2 and exploited by Wehausen 3 yields the deterministic evaluation of a vehicle's reponse to a specified time history of the sea surface. This approach was extended by Henry 4 to the prediction of the rigid body and elastic response of hydrofoil craft. Either method has advantages in attaining effective designs for marine vehicles. This paper is concerned with the deterministic approach and, in particular, with the requirement that the impulse response functions satisfy the law of causality, which states that no physical system can respond to future causes, i.e., the cause must always precede the effect. The deterministic approach to calculating response of physical systems has been highly developed in the field of control theory5 which deals with systems described by ordinary differential equations, i.e., with inputs and outputs functions of one independent variable, time. For stable, linear, timeinvariant systems, it is well known that the output y(t) of a so-called dynamical system can be predicted from the previous history of the input rj(r), T < t by the convolution integral