Vortices in oscillating liquid

In a paper “on the Circulation of Air observed in Kundt’s Tubes, and on some Allied Acoustical Problems,” I applied the equations of viscous incompressible fluid to show that the effect of the bottom of the containing vessel was to generate permanent vortices in the vibrating fluid. It was remarkable that the intensity of the vortical motion, when fully established, proved to be independent of the magnitude of the viscosity, so that the effects could not be eliminated by merely supposing the viscosity to become extremely small. The expression found for the vortices was simple. The horizontal componentu of the primary motion near the bottom beingu =u 0 coskx cosnt , the component velocities of the vortical motion areu' = 3/8u 0 2 sin 2kx /V e-2ky (1-2ky ), –v' = 3/8u 0 2 cos 2kx /V e-2ky 2ky ,y being measured upwards from the bottom, and V (=n /k ) the velocity of propagation of waves of the length in question. According to these expressions, the vortical motion is downwards over the places whereu has its greatest alternating values. In the case of water contained in a tank and vibrating in its simplest mode, the theoretical motion is downwards in the middle and upwards at the ends. To guard against misinterpretation, it may be well to add that quite close to the bottom the motion, as calculated, is of a quite different character. In a recent paper, Mrs. Ayrton has examined, with much experimental skill, the vortices arising when water oscillates in a narrow tank, and has obtained results which differ somewhat widely from what are indicated in the above formulæ. Near the bottom, and especially when the depth is small, there are indeed vortices of this character; but, in general, the most conspicuous feature consists of vortices revolving in the opposite direction, the water rising in the middle of the tank and falling at the ends. The first thought that occurred to me was that Mrs. Ayrton’s vortices might be due to defect of freedom in the surface, such as might be supposed to arise from a greasy film opposing extensions and contractions; but in some experiments that I tried, the vortical motion did not seem to be much influenced by cleansing the surface, and the question was suggested as to whether the free surface itself might not originate vortices in somewhat the same way as the bottom does, and more potently on account of the greater velocities of the primary motion there prevailing. I do not remember whether I had any clear view on this question when I wrote the former paper. Vortices originating other­wise than at the bottom were ignored, but I may not have intended to exclude their possibility.