the presence of a wind drift layer with the surface drift velocity (relative to the fluid at infinite depth) in the same direction as the wave phase speed, the incipient breaking condition occurs at smaller wave amplitudes than predicted by Stokes:

(2)

where q=(U−U(0))/U, where U(0) is the fluid velocity at the water surface in the reference frame of the wave crest but with no wave present. The result of Stokes is reproduced when q=0. For the case of ship waves, the wind drift layer would be replaced by the wake from an upstream breaker or the ship hull.

Computations of waves propagating in the presence of free surface shear layers have been reported by Simmen and Saffman [7] and Teles Da Silva and Peregrine [8]. In Simmen and Saffman [7], waves on a fluid with constant vorticity and infinite depth were considered, while in Teles Da Silva and Peregrine [8] waves on a layer of fluid with constant vorticity and finite depth were considered. In both cases wave profiles, extreme wave heights and wave propagation speeds were presented.

Experimental studies of the incipient breaking conditions for steady two-dimensional waves generated in calm water have been reported by Salvesen and von Kerczek [9] and Duncan [3]. In both studies, the waves were generated with a submerged hydrofoil moving at constant speed, depth, and angle of attack in a towing tank. In Salvesen and von Kerczek [9], the incipient breaking condition was determined by fixing the depth and angle of attack of the foil and varying the foil speed from one experimental run to another. At low speeds, the wave steepness was small and no breaking occurred. As the speed was increased, the wave became steeper and, for a high enough speed, the wave broke. If the foil speed was increased past this point, the breaking eventually stopped. Thus, speeds just less than the speed for which breaking started and just high enough for breaking to stop were chosen, and the wave slope was measured at each of these incipient breaking conditions. This procedure was repeated for several depths of submergence. The maximum surface slope of these incipient breaking waves varied from 11 to 25 degrees and did not show any consistent trend. Duncan [3] found that for fairly steep nonbreaking waves, breaking could be triggered by dragging a cloth for 1 or 2 seconds on the water surface ahead of the wave. For small enough wave steepnesses, when the cloth was removed, the wave would stop breaking. However, for higher wave steepnesses the wave would continue to break after the cloth was removed. The wave profiles measured at the incipient breaking condition determined by whether or not the wave would continue breaking when the cloth was removed were very consistent. The maximum slope of each profile was found to be about 16°; this value increased slowly with towing speed. Even though the cloth was used momentarily to trigger breaking, the above defined incipient breaking condition is for a wave in calm water.

In the present paper, the effect of a steady surface wake on the incipient breaking condition of a steady wave is examined experimentally and numerically. In the experiments, a plastic sheet is dragged along the water surface at a fixed distance ahead of the steady wave created by a towed hydrofoil. Unlike the experiments of Duncan [3], in the present experiments the plastic sheet was always present in front of the wave. With the hydrofoil at a fixed depth of submergence (one for which it produces a nonbreaking wave in calm water), the distance, Δx, between the trailing edge of the plastic sheet and the hydrofoil was varied to obtain the incipient breaking condition. For small Δx, the local surface drift near the wave crest, q, is high and the wave tends to break even when its amplitude is small. For large Δx, q is small and the wave does not break, as if it were propagating in calm water. The incipient breaking wave was taken as the nonbreaking wave for which breaking will start if Δx is decreased by a small amount. Wave profile measurements are taken at the incipient breaking conditions and the wakes of the plastic sheets are characterized through measurements of the mean horizontal velocity distributions. These measurements are used to quantify the effect of q and the wake momentum thickness on the incipient breaking conditions. Direct numerical simulations of a similar flow are performed using a fully nonlinear, inviscid, two-dimensional, free-surface flow code for incipient breaking waves, and large wave simulations for breaking waves. The incipient breaking conditions found in the experiments are compared to the theory of Banner and Phillips [6] and to the results of the numerical simulations. The experimental data and the numerical results are further used to explore the physics of the instability processes at the incipient breaking condition.

The remainder of this paper is divided into five sections. In Section 2, the details of the experimental setup and measurement techniques are presented. This is followed in Section 3 by a description of the experimental results. In Section 4, the numerical model is presented along with some typical results. The experimental and numerical results are compared and discussed in Section 5. Finally, the conclusions are presented in Section 6.



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