Dispersion Relation in the Inertial Regime

Note also that a small drop vibrating in air has its wavelength necessarily fixed by its radius R, Since we are then in the capillary limit ( k), equation (5.54) enables us to deduce the dimensional form of the period r of oscillation. The result is r oc pR / y, For a millimeter-size drop, this period is of the order of a millisecond. 

Equation (5.54) provides the phase velocity c = ui/qoi the surface waves. We find 

A surface excited by a tuning fork vibrating at a frequency uj/2t = 440 Hz generates capillary waves with a wavelength of the order of a millimeter [equation (5.54)] traveling at a velocity of about 60 cm/s [equation (5.55)]. A remarkable property predicted by equation (5.55) is the existence of a minimum for c. The minimum c is obtained when A = y/2 gX/The result is 23 cm/s for water. 

A fisherman standing on the edge of a fast-moving river can see around and downstream of his line a remarkable capillary wake that has fascinated physicists (such as Lord Rayleigh, Lamb, Lighthill, and others) for a cen-tury.30,31 rj Yie wake forms only when the speed of the river relative to the line exceeds the minimum velocity c of the associated waves. Generally speaking, the entire structure of the wake can be analyzed in detail with the help of equation (5.55). The interested reader may want to catch up with recently published developments on this topic. 

All these results must of course be modified if the fluid has a thickness /i of the order of or less than the wavelength A. An example is ocean waves washing up on a beach. If the effect of viscosity can still be neglected, it is necessary to introduce an additional boundary condition, namely, that the vertical component of the velocity Vz be zero at 2 = —/i. The result when qh is small is 

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