Tube Waves in A Borehole
Tube Waves in A Borehole : In an empty cylindrical hole, a kind of surface wave can propagate along the axis of the hole with energy confined to the vicinity of the hole. They exhibit dispersion with phase velocity increasing with the wavelength. At wavelengths much shorter than the bore hole radius, they approach Rayleigh waves. The phase velocity reaches the shear velocity at wavelengths of about three times the radius. Beyond this cutoff wavelength, they attenuate quickly by radiating S waves. In a fluid-filled cylindrical hole, in addition to a series of multi-reflected conical waves propagating in the fluid, tube waves exist without a cutoff for the entire period range. At short wavelengths, they approach Stoneley waves for the plane liquid-solid interface. For wavelengths longer than about 10 times the hole radius, the velocity of tube waves becomes constant, given in terms of the bulk modulus ? of the fluid and the rigidity ? of the solid, by ? = c / ?(1 + ?/?), where c is the acoustic velocity in the fluid
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