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Capillary waves have short wavelengths. Both the phase and group velocities increase as wavelengths become shorter and group velocity is greater than phase velocity. Dissipation of the waves by molecular viscosity is very rapid. The characteristic shape of the water surface, sinusoidal for small amplitudes, becomes distorted with more sharply curved troughs than crests as wave amplitude increases. In all these respects, ripples contrast sharply with gravity waves. Mathematical formulas have been derived that relate the phase velocity, group velocity, and frequency to the wave- length of low-amplitude waves on still water in the absence of wind. In such waves, gravity and surface tension play equal roles in the restoring force; longer waves are considered to be gravity waves and shorter are capillary waves. In nature, ripples are observed to grow rapidly when the wind blows and to die away rapidly when the wind stops. When the water surface is uncontaminated, ripples die away to 37% of their original amplitude in a period of time that is related to the wavelength and the kinematic viscosity of water. For ex- ample, for a wave with a wavelength of 17 mm and a kinematic velocity of 10 m/s, the period of time is 3.8 s. When the water surface is contaminated, as by an oil film or other surface-active agent, ripples are damped still more rapidly, because the con- taminated surface tends to act as an inextensible film against which the water motions due to the ripples must rub. In such a case for the example given, the period of time becomes 0.86 s. Under low-wind conditions, this increased damping al- most completely inhibits ripple growth: the surface appears glassy smooth and is called a slick. It has been observed that even short gravity waves grow at an inappreciable rate under such conditions; the interpretation here is that the fine scale of roughness presented to the wind by a rippled surface is involved in the formation and growth of gravity waves. How- ever, ripples have been observed to form on clean water in the absence of wind by nonlinear processes occurring at the sharp crests of short, steep gravity waves; consequently the formation of both gravity waves and ripples is an interconnected process. Ripple trains formed under low-wind conditions derive their energy at the expense of gravity waves and are called parasitic capillaries. In this case, the capillary wave train is propagating in a moving stream of water, the orbital current of the underly- ing gravity wave. The longer wave parts of the capillary train would have a lower phase velocity in still water than either the shorter wavelength parts of the train or of the gravity wave, but they maintain their position relative to the crest of the gravity wave because they ride in a favorable part of the orbital cur- rent of the gravity wave. Short capillaries, having high phase velocity, do not need this aid to keep in phase and their higher group velocity enables them to proceed to a leading position where the gravity-wave orbital velocity vanishes or even op- poses their motion. Internal waves Internal waves are wave motions of stably stratified fluids in which the maximum vertical motion takes place below the surface of the fluid. The restoring force is mainly due to gravity; when light fluid from upper layers is depressed into the heavy lower layers, buoyancy forces tend to return the layers to their equilibrium positions. Internal waves have been found in the atmosphere as lee waves (waves in the wind stream downwind from a mountain) and as waves propagated along an inversion layer (a layer of very stable air). They are also associated with wind shears at the lower boundary of the jet stream. In the oceans, internal oscillations have been observed wherever suit- able measurements have been made. The observed oscillations can be analyzed into a spectrum with periods ranging from a few minutes to days. At a number of locations in the oceans, internal tides, or internal waves having the same periodicity as oceanic tides, are prominent. Internal waves are important to the economy of the sea because they provide one of the few processes that can redis- tribute kinetic energy from near the surface to abyssal depths. When they break, they can cause turbulent mixing despite the normally stable density gradient in the ocean. Internal waves are known to cause time-varying refraction of acoustic waves because the sound velocity profile in the ocean is distorted by the vertical motions of internal waves. The result is that quasi- horizontal propagation of sound shows phase incoherence and large changes of intensity with time at ranges where the refrac- tion has led to divergence or convergence of rays. The vertical distribution of motions and phase velocity of internal waves depends on the vertical gradient of density in the fluid and the frequency of the generating forces. There is a simple density distribution that is illustrative: The fluid consists of two homogeneous layers, a lighter one on top of a heavier one, such as kerosene over water. The internal waves in this system are sometimes called boundary waves, because the maximum vertical motion occurs at the discontinuity of density at the boundary between the two fluids. Internal waves move at a slow speed, of the order of a few knots in the deep oceans. The effect of the rotation of the Earth is to increase the phase velocity of waves having periods long enough to approach one pendulum day. When there is a continuous distribution of density in a fluid, as in the ocean or atmosphere, internal waves are possible only for frequencies that are lower than the maximum value given by a mathematical relationship known as the Väisälä-Brunt frequency, which is related to the downward rate of increase of the density and the velocity of sound in the fluid. In the ocean this maximum frequency occurs in the thermocline, where it commonly amounts to about one-fifth cycle per minute. At any frequency lower than this limit, there is an infinity of possible modes of internal waves. In the first mode, the vertical motion Ocean Waves (continued) + ward ' s science