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The stability of the plane free surface of a liquid in vertical periodic motion
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1954
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A vertically vibrating vessel containing a heavy liquid can produce standing waves or maintain a plane free surface depending on vibration frequency and amplitude. This study theoretically investigates the stability of a plane free surface in a vertical cylindrical vessel with a horizontal base, assuming an ideal frictionless fluid that meets the walls at 90°. The authors construct a stability chart that links cylinder cross‑section, vibration frequency, and amplitude to mode excitation, predicting that the m‑th mode appears when its parameter pair lies in an unstable region and that the surface remains stable if no modes are excited, and they confirm the theory experimentally with water. The theory predicts when standing.
A vessel containing a heavy liquid vibrates vertically with constant frequency and amplitude. It has been observed that for some combinations of frequency and amplitude standing waves are formed at the free surface of the liquid, while for other combinations the free surface remains plane. In this paper the stability of the plane free surface is investigated theoretically when the vessel is a vertical cylinder with a horizontal base, and the liquid is an ideal frictionless fluid making a constant angle of contact of 90° with the walls of the vessel. When the cross-section of the cylinder and the frequency and amplitude of vibration of the vessel are prescribed, the theory predicts that the m th mode will be excited when the corresponding pair of parameters (p m , q m ) lies in an unstable region of the stability chart; the surface is stable if none of the modes is excited. (The corresponding frequencies are also shown on the chart.) The theory explains the disagreement between the experiments of Faraday and Rayleigh on the one hand, and of Matthiessen on the other. An experiment was made to check the application of the theory to a real fluid (water). The agreement was satisfactory; the small discrepancy is ascribed to wetting effects for which no theoretical estimate could be given.