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Interaction of waves and a shear flow Hughes, Blith Alvin

Abstract

A series of experiments has been undertaken in which three major properties of a surface tension-gravity wave system have been critically examined. The results of these experiments have been compared with existing theories. The three properties are: viscous decay in the absence of mean flow, which has been compared to the theory given in Lamb (1932) #348; propagation velocities in the presence and absence of mean flow, compared to Lamb (1932) #267, and the change of wave energy on crossing a stable Couette shear flow, compared to two theories - one obtained from the Navier-Stokes equations including terms up to second order in wave slope; the other, following previous authors, assuming any direct interaction of the waves with the shear flow to be negligible. According to the theory obtained from the Navier-Stokes equations the divergence of the rate of transport of surface wave energy is equal to the rate of change of wave energy due to the interaction of the mean flow and the wave system plus the rate of change of energy due to viscous decay. An optical system was used to measure the maximum wave slope, the wavenumbers and the shear velocities. A grid of point source lights was set up to reflect off a previously chosen part of the shear flow into a properly oriented camera. A series of pulses of waves, generated by an electromechanical transducer, were then sent across this region of the flow. For each pulse of waves a photograph was taken of the oscillating images of the lights with the exposure time longer than one wave period. The resulting streaks on the film are proportional to the maximum wave slopes at the positions from which the undeviated light reflects. A series of parallel straight white strings were photographed with the aid of a flash unit when waves crossed the flow. This was then used to determine lines of constant phase from which the wavelength and hence the wavenumber was measured at various positions across the flow. The viscous decay and part of the propagation measurements were obtained in this way but with no mean flow. Results indicate that an anomalous region of wave properties exists for wavenumbers near 2.7 cm. ̄¹. For a set of data in which the wavenumbers were always less than 1.8 cm.̄¹, it was found that the viscous decay rate and the propagation laws agree with theory to within the experimental error, and the interaction measurements fit the theory with the non-linear term included rather than the traditional theories.

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