Shear wave propagation

In the methods of the preceding section, the propagation of a shear wave into the viscoelastic liquid is not observed directly only its effect on the propagating surface is detected and measured. On the other hand, if the damping is not too severe, so that a wave train of several maxima and minima is perceptible, the wavelength and attenuation can be measured directly, as described in this section. The sample must be sufficiently large so that the waves are attenuated before they can experience reflection. 

For a shear wave traveling under conditions where it approximates a one-dimensional disturbance in the x direction, the damping is exponential and the amplitude (shear displacement, u) can be represented as 

Exponentially damped shear wave with definitions of characteristic parameters (from reference 116). 

Shear wave propagation in a rectangular vessel (from reference 114). 

From measurements of the wavelength and attenuation the components of the complex shear modulus can be calculated by the following equations  

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J. Bishop, G. Poole, M. Leitch and D. B. Plewes, Magnetic resonance imaging of shear wave propagation in excised tissue, J. Magn. Reson. Imaging, 1998, 8, 1257-1265. 

Fig. 11.4. Velocities of bulk and surface waves in an (001) plane the angle of propagation in the plane is relative to a [100] direction, (a) Zirconia, anisotropy factor Aan = 0.36 (b) gallium arsenide, anisotropy factor Aan = 1.83 material constants taken from Table 11.3. Bulk polarizations L, longitudinal SV, shear vertical, polarized normal to the (001) plane SH, shear horizontal, polarized in the (001) plane. Surface modes R, Rayleigh, slower than any bulk wave in that propagation direction PS, pseudo-surface wave, faster than one polarization of bulk shear wave propagating in...

J.S. Lee and G.G. Fuller, The spatial development of transient couette flow and shear wave propagation in polymeric liquids by flow birefringence, J. of Non Newt. Fluid Mech., 26, 57 (1987). 

J. S. Lee and G. G. Fuller, Shear wave propagation in polymer solutions following a step increase of shear rate, J. Non-Newt. Fluid Mech., 39, 1 (1991). 

Example 2.5 Calculate the attenuation for a y-polarized shear wave propagating along the x axis of a cubic crystal, based on the elastic constitutive relation modified to include viscous damping. Equation 2.17. 

In order to satisfy the stress-free boundary condition, coupled compressional and shear waves propagate together in a SAW such that surface traction forces are zero (i.e., T y = 0, where y is normal to the device surface). The generalized surface acoustic wave, propagating in the z-direction, has a displacement profile u(y) that varies with depth y into the crystal as... 

Most applications of US-detection in liquids use compressive, longitudinal waves shear waves propagate over macroscopic distances in solids, but not in liquids, and other modes of US propagation are of little use with liquid samples. 

When the shear waves propagate through the elastic layer, or the elastic plate and reach the steady state, the type of the wave, SH wave for example, and it s dispersion relation are determined by the boundary conditions at the plate surfaces [7]. We have assumed that the sound waves modulate the stress fields at the tip of the crack, and then solved the wave equations with the boundary conditions at the surfaces of the crack and the plate. If the analysis is extended to derive the higher order fields and the dispersion relation of the wave is then obtained, such a wave do exist in the steady state. In this case we could confirm the existence of such "new wave" associated with the crack. Much algebra is required to obtain the higher order fields, however, it is not difficult to see the structure of the fields with the boundary condition at the plate surfaces. We find the boundary conditions at the plate surfaces for the second order stress fields are satisfied by the factor, cos /5 z, in the similar manner to Eq. 

NoNaq"). The shear wave transducer for this sample was a 10 MHz lithium niobate transducer bonded with a cyanoacrylate polyester. For the lower density sample both PZT and quartz transducers operating at nominally 3 to 4 MHz were used for both longitudinal and shear wave propagation. They were bonded with either an alcohol-glycerine-based couplent or phenyl salicylate. The detected ultrasonic echoes were rectified and displayed on an oscilloscope equipped with time-delay circuitry for the transit time measurements. 

When the resonant condition of a thickness-shear oscillation is satisfied for AT-cut piezoelectric crystals, a shear wave propagates through the bulk of the material, perpendicular to the faces of the crystals. The fundamental resonance frequency, fo is given by ... 

This chapter is concerned with just one of this class of thickeners, namely a hydrophobically modified (hydroxyethyl)cellulose (HMHEC), Hercules WSP D-47. Gelman and Barth (i) reported on the viscosity of such HMHECs. This sample contained up to four hexadecyl chains grafted to the cellulose backbone. The preparative route was that described by Landoll (2). The dilute solution properties were characterized by capillary viscom-etry, whereas the more concentrated solutions were characterized by continuous-shear viscometry, forced oscillation measurements, and shear-wave propagation. In addition, the adsorption onto polymer latex particles was investigated. 

Shear Wave Propagation. A pulse shearometer (Rank Bros.) was used to measure the propagation velocity of a shear wave through the weak gels formed by the solutions of HMHEC in dilute NaCl. The polymer concentration range studied was 0.5-2.0%. With this apparatus, the frequency of the shear wave is approximately 1200 rad s" and the strain is <10 . At this strain, n pst systems behave in a linear viscoelastic fashion, and the wave-rigidity modulus, G is... 

Cavic, B. A., Thompson, M. (2002). Interfacial nucleic acid chemistry studied by acoustic shear wave propagation. Anal ChimActa 469, 101-113. 

In plane-harmonic shear-wave propagation in a linear medium of density p, G and G" are given by,... 

Velocity of a shear wave propagated perpendicular to the c-axis with polarization also perpendicular to the... 

Dcs = shear wave propagated parallel to the unique axis with any polarization direction appropriate to a shear wave. 

In zero magnetic field their effect just cancels the contribution of the first-order rotational interaction but as pointed out before they may become important for if 7 0. In zero field the velocity of a shear wave propagating in z-direction and polarized in x-direction is simply... 

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