On the theory of surface waves in water generated by moving disturbances

Abstract

The wave profile generated by an obstacle moving at constant veiocity U over a water surface of infinite extent appears to be stationary with respect to the moving body provided, of course, the motion has been maintained for a long time. When the gravitational and capillary effects are both taken into account, the surface waves so generated may possess a minimum phase velocity c[sub]m characterized by a certain wave length, say [lambda][sub]m (see Ref. 1, p. 459). If the velocity U of the solid body is greater than c[sub]m, then the physically correct solution of this two-dimensional problem requires that the gravity waves (of wave length greater than [lambda][sub]m) should exist only on the downstream side and the capillary waves (of wave length less than [lambda][sub]m) only on the upstream side. If one follows strictly the so-called steady-state formulation so that the time does not appear in the problem, one finds in general that it is not possible to characterize uniquely the mathematical solution with the desired physical properties by imposing only the boundedness conditions at infinity. [Footnote: In the case of a three-dimensional steady-state problem, even the condition that the disturbance should vanish at infinity is not sufficient to characterize the unique solution.] Some stronger radiation conditions are actually necessary. In the linearized treatment of this stationary problem, several methods have been employed, most of which are aimed at obtaining the correct solution by introducing some artificial device, either of a mathematical or physical nature. One of these methods widely used was due to Rayleigh, and was further discussed by Lamb. In the analysis of this problem Rayleigh introduced a "small dissipative force", proportional to the velocity relative to the moving stream. This "law" of friction does not originate from viscosity and is hence physically fictitious, for in the final result this dissipation factor is made to vanish eventually. In the present investigation, Rayleigh's friction coefficient is shown to correspond roughly to a time convergence factor for obtaining the steady-state solution from an initial value problem. (It is not a space-limit factor for fixing the boundary conditions at space infinity, as has usually been assumed in explanation of its effect). Thus, the introduction of Rayleigh's coefficient is only a mathematical device to render the steady-state solution mathematically determinate and physically acceptable. For a physical understanding, however, it is confusing and even misleading; for example, in an unsteady flow case it leads to an incomplete solution, as has been shown by Green. Another approach, purely of a physical nature, was used by Michell in his treatment of the velocity potential for thin ships. To make the problem determinate, he chose the solution which represents the gravity waves propagating only downstream and discarded the part corresponding to the waves traveling upstream. For two-dimensional problems with the capillary effect, this method would mean a superposition of simple waves so as to make the solution physically correct. Some other methods appear to be limited in the necessity of interpreting the principal value of a certain kind of improper integral. In short, as to their physical soundness and mathematical rigor, or even to their merits or demerits, the preference of one method over the others has remained nevertheless a matter of considerable dispute. Only until recently the steady-state problem has been treated by first formulating a corresponding initial value problem. A brief historical sketch of these methods is given in the next section. The purpose of this paper is to try to understand the physical mechanism underlying the steady configuration of the surface wave phenomena and to clarify to a certain extent the background of the artifices adopted for solution of steady-state problems. The point of view to be presented here is that this problem should be formulated first as an initial value problem (for example, the body starts to move with constant velocity at a certain time instant), and then the stationary state is sought by passing to the limit as the time tends to infinity. If at any finite time instant the boundary condition that the disturbance vanishes at infinity (because of the finite wave velocity) is imposed, then the limiting solution as the time tends to infinity is determinate and bears automatically the desired physical properties. Also, from the integral representation of the linearized solution, the asymptotic behavior of the wave form for large time is derived in detail, showing the distribution of the wave trains in space. This asymptotic solution exhibits an interesting picture which reveals how the dispersion* generates two monochromatic wave trains, with the capillary wave in front of, and the gravity wave behind, the surface pressure. *[Footnote: By dispersive medium is meant one in which the wave velocity of a propagating wave depends on the wave length, so that a number of wave trains of different wave lengths tends to form groups, propagating with group velocities which are in general different from the phase velocities of individual wave trains. In case of waves on the water surface, both the gravity and surface tension are responsible for dispersion.] The special cases U< c[sub]m and U = c[sub]m are also discussed. The viscous effect and the effect of superposition are commented upon later. Through this detailed investigation it is found that the dispersive effect, not the viscous effect plays the significant role in producing the final stationary wave configuration.

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