Bulk Wave Propagation

In the compound systems described above, the polymer is subjected to com-pressional stress only, with no shear components, because the confining liquid transmits no perceptible shear stresses. If, however, longitudinal waves are propagated in a continuous specimen of polymer with at least one dimension large compared with the wavelength, the sample experiences both bulk and shear stresses and deformations which are combined in a manner dependent on the sample shape. 

at the other extreme, the sample dimension normal to the stress is large compared with the wavelength, the situation becomes relatively simple again and the wave propagation is governed by the complex piodulus M  

If the components of G have been determined in a separate experiment, those oi K can be obtained from M by difference.  

For a soft solid, in a frequency range where G K, M is practically equal to K and no subtraction is necessary but and G may be of comparable magnitude. 

Since the subtractions indicated in equations 2 and 3 may cause loss of predsion, it is desirable to measure both M and G in the same experiment with as little manipulation of the apparatus as possible. This can be achieved by propagating both longitudinal and transverse waves through the same sample the velodty and attenuation of the latter provide G and G in accordance with equations 39 and 40 of Chapter 5. Methods for these measurements have been described by Brad-field, Wada,, Kono, and Waterman. In the latter two, the sample in the form of a plate is placed between a transmitting and a receiving transducer as in Fig. 8-2 

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This technique was used successfully to show that bovine plexiform bone was definitely orthotropic while bovine haversian bone could be treated as transversely isotropic [Lipson and Katz, 1984]. The results were subsequently confirmed using bulk wave propagation techniques with considerable redundancy [Maharidge, 1984]. 

Equation (14.1 la) has a wave vector of different amplitude and represents a compressional or longitudinal wave. In this case particle motion is parallel to the direction of wave propagation. The wave vector has a different value as a consequence of the fact that a different stiffness constant applies to compressive particle motion, and therefore the wave will travel with a different phase velocity. Thus, in isotropic solids three modes of bulk wave propagation exist of which two are degenerate and can only be formally distinguished by their polarization. 

FIG.8-2. Apparatus for longitudinal bulk wave propagation measurements, by echoing longitudinal pulses through a liquid with and without a polymeric sample in the path. (Nolle and Mowry.p ... 

FIG.8-4. Apparatus for longitudinal bulk wave propagation measurements in a viscoelastic liquid, with variable path length. (Mason, Baker, McSkimin, and Heiss. )... 

The elastic properties of PS depend on its microstructure and porosity. The Young s modulus for meso PS, as measured by X-ray diffraction (XRD) [Ba8], acoustic wave propagation [Da5], nanoindentation [Bel3] and Brillouin spectroscopy [An2], shows a roughly (1-p)2 dependence. For the same values of porosity (70%), micro PS shows a significantly lower Young s modulus (2.4 GPa) than meso PS (12 GPa). The Poisson ratio for meso PS (0.09 for p=54%) is found to be much smaller than the value for bulk silicon (0.26) [Ba8]. 

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...

Fig. 4.19 Lattice distortion in Rayleigh wave defining the saggital plane (a) longitudinal (bulk) wave (b) y-polarized (shear bulk) wave (c) y-0-z polarized Rayleigh waves (SAW waves) all propagating in the z-direction...

Three groups of phenomena affect the frequency-dependence of ultrasonic wave propagation classical processes, relaxation, and scattering, of which scattering is likely to dominate in foodstuffs due to their particulate nature. The two classical thermal processes are radiation and conduction of heat away from regions of the material, which are locally compressed due to the passage of a wave they can lead to attenuation but the effect is negligible in liquid materials (Herzfield and Litovitz, 1959 Bhatia, 1967). The third classical process is due to shear and bulk viscosity effects. Attenuation in water approximates to a dependence on the square of the frequency and because of this it is common to express the attenuation in more complex liquids as a()/o or a(f)jf2 in order to detect, or differentiate from, water-like properties. 

The use of heterodyne detection to monitor vibrational oscillations even after traveling wave propagation out of the excitation region has also been demonstrated in the case of acoustic modes in bulk and thin... 

Figure 2.1 Pictorial representations of elastic waves in solids. Motions of groups of atoms ate depicted in these cross-sectional views of plane elastic waves propagating to the right. Vertical and horizontal displacements are exaggerated for clarity. Typical wave speeds, Vp, are shown below each sketch, (a) Bulk longitudinal (compressional) wave in unbounded solid, (b) Bulk transverse (shear) wave in unbounded solid, (c) Surface acoustic wave (SAW) in semi-infinite solid, where wave motion extends below the surface to a depth of about one wavelength, (d) Waves in thin solid plates.

Figure 3.39 (page 114) shows the phase velocities of the waves as a function of the product k4, where k, = 27t/A, A, is the wavelength of the bulk transverse (shear) wave in the medium of which the plate is made, and d is the plate thickness. The waves divide naturally into two sets symmetric waves (denoted by So, S],. ..) whose particle displacements are symmetric about the neutral plane of the plate, and antisymmetric waves (Aq, A, . ..), whose displacements have odd symmetry about the neutral plane. Figure 3.38 shows that for sufficiently thin plates (M < 1-6), only two waves exist — the lowest-order symmetric mode (Sq) and the lowest-order antisymmetric mode (Aq). These are the modes shown earlier in Figure 2.0d. The plate mode that we will emphasize here is the Ao mode, in which the elements of the plate undergo flexure as the wave propagates. The shape of a plate during propagation of this flexural mode has been likened to that of a flag waving in the wind.

Chugging in liquid-propellant rockets is only one of many examples of system instabilities in combustion devices [135], [136]. As may be seen from the contribution of Putnam to [137], a considerable amount of research has been performed on mechanisms of these instabilities. Interactions of processes occurring in intakes and exhausts with those occurring in the combustion chamber typically are involved, and it may or may not be necessary to consider acoustic wave propagation in one or more of these components in theoretical analyses [138]-[142]. Here we shall not address problems involving acoustic wave propagation we shall restrict our attention to bulk modes, in which spatial variations of the pressure in the combustion chamber are negligible. 

The effective P may be determined with the electron beam apparatus. When the sample (slab geometry) is thick enough to absorb all of the incident electrons, a compressive stress wave propagates from the irradiated region into the sample bulk. A transducer, located just beyond the deposition depth, may be used to record the stress pulse. Alternatively, the displacement or velocity of the rear surface of sample may be observed optically and used to infer the initial pressure distribution from the experimentally measured stress history. Knowledge of the energy-deposition profile then permits the determination of the Gruneisen coefficient. 

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