The frequency equation for longitudinal waves in a viscoelastic tube of infinite length filled with a viscous fluid is solved for the complex wavenumber, k(=k1+ik2). Thus for two modes, Young's mode and Lamb's mode, an explicit expression for k is obtained which yields the phase velocity c(=ω/k1, ω=radial frequency) and the damping constant k2. The formulas for k, c, and k2 are valid for wall thicknesses up to the size of the inner radius of the tube. The approximation as compared with results obtained numerically with an IBM 7040 digital computer from the original frequency equation involves an error which in almost all cases is considerably less than 1.5%. For Lamb's mode, as it was found earlier for the torsional mode, an increase of the fluid viscosity increases the damping at the higher frequencies but decreases the damping at the lower frequencies. The theoretical results are compared with those of experiments in which the longitudinal waves are generated in the fluid and the fluid in its turn moves the wall. The experimental curves thus obtained show the pattern of Young waves, and the agreement with the Young curves obtained from the formulas mentioned above is satisfactory. For torsional waves, formulas for k, c, and k2 analogous to these for longitudinal waves are given.

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Articles you may be interested inA long-range-corrected density functional that performs well for both ground-state properties and timedependent density functional theory excitation energies, including charge-transfer excited states Development of a sum-over-states density functional theory for both electric and magnetic static response properties Not only for the the heart, but also for papillary muscle a contraction variable K can be defined (in analogy with the deformation variable A ) in a phenomenological approach that makes use of the finding that two of the three constants (c 2 and c 3 ) in the isochronics of such a muscle can be considered functions of the third one (K). This fact provides four constants (k i' k2' k~, and k 4 ) of an energy density function. A tentative bridge to the heart (for clinical application) is laid; four analogous constants of the same order of magnitude can be measured there.

The time dependence of the contraction variable (Qc) for dog hearts was found to be largely independent of preload and afterload. This appears to make its clinical determination by pressure-volume measurements not unfeasible and could have important diagnostic consequences. The independence of the function Qc=?c(t) on the deformation variable (Q) led also to the derivation of some thermodynamic formulas for this contractile material.

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