The transient behavior of small‐amplitude standing waves on the plane surface of an infinitely deep viscous liquid is investigated. An integro‐differential equation of motion for the free surface is derived and solved exactly. The small‐time behavior of the solution agrees with that computed in the approximation of irrotational flow, and the large‐time one with the results for the discrete spectrum obtained by means of the standard normal‐mode analysis. In between these two asymptotic regimes, however, the exact solution is significantly different from either approximation, except when the effect of viscosity is very small.

H. Lamb, Hydrodynamics (Cambridge U.P., Cambridge, England, 1932; reprinted by Dover, New York, New York, 1945), 6th ed.
G. K. Batchelor, An Introduction to Fluid Dynamics (Cambridge U.P., Cambridge, England, 1970), p. 364.
A. B. Bassett, A Treatise on Hydrodynamics (Deighton, Bell and Co., London, 1888; reprinted by Dover, New York, 1961), Vol. 2, p. 309.
S. Chandrasekhar, Hydrodynamic and Hydromagnetic Stability (Clarendon Press, Oxford, England, 1961), p. 451.
A. Prosperetti, in Proceedings of the International Colloquium on Drops and Bubbles, edited by D. J. Collins (Jet Propulsion Laboratory, Pasadena, California, 1975).
L. D. Landau and E. M. Lifshitz, Fluid Mechanics (Addison‐Wesley, Reading, Mass., 1959), Sec. 12.
H. L. Dryden, F. P. Murnaghan, and H. Bateman, Hydrodynamics (Dover, New York, 1956), p. 188.
W. Gautschi, in Handbook of Mathematical Functions, edited by M. Abramowitz and I. A. Stegun, National Bureau of Standards Applied Mathematics Series, No. 55 (U.S. Government Printing Office, Washington, D.C., 1964), p. 301.
A. Erdelyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi, Tables of Integral Transforms (McGraw‐Hill, New York, 1954), Vol. 1.
D. V. Widder, The Laplace Transform (Princeton U.P., Princeton, New Jersey, 1941), Chap. 5.
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