Lamb’s solution

The responses in Fig. 3.8 are calibrated results by two methods of the sensor calibration. One is a calibration method by NIST, as illustrated in Fig. 3.10 (Breckenridge 1982). A large steel block of 90 cm diameter and 43 cm deep was employed. As a step-function impulse, a glass capillary source was employed, and elastic waves were detected by a capacitive transducer and by a sensor under test. The calibration curve was obtained as a ratio of the response of the sensor to that of the eapacitive transducer. The capacitive transducer (sensor) could record a Lamb s solution due to surface pulse as discussed in Chapter 7. It is reasonably assumed that the capacitive transducer detect the vertical displacement at the surface due to a step-function force. The other is known as a reciprocity method, which Hatano and Watanabe (Hatano Watanabe 1997) suggested to use, and confirmed an agreement with the NIST methods. As seen in Fig. 3.8, it is demonstrated that both method can provide similar calibration curves. 

In Fig. 3.11, velocity motion detected by the laser vibrometer is shown at the middle and that of Lamb s solution is shown at the bottom. Remaika-ble agreement is confirmed except the latter reflections in the detected wave. Because a concrete block was employed in the experiment, these reflected motions are observed. At the top, frequency spectrum of the detected wave is shown as a solid curve and is compared with that of Lamb s solution denoted by a broken curve. Reasonable agreement is again confirmed. This result suggests an application of the laser vibrometer for the absolute calibration instead of the capacitive transducer. 

In elastodynamics as well as in seismology, the problem where a force is applied in a half space is called Lamb s problem. This is because the problem was first solved by Lamb [1904]. Then, Pekeris published famous results of Lamb s solutions due to a surface pulse (Pekeris 1955) and a buried pulse (Pekeris 1955). 

A generalized solution suitable for numerical computation was comprehensively reported by Johnson [1974], The computational code is already available in the literature (Ohtsu 1984). Accordingly, Green s functions in a half space are normally called Lamb s solutions. In most cases, the solutions due to step-function are presented as formulated in Eq. 7.13. 

The configuration of the detection is illustrated in Fig. 7.1, showing two cases. One is the case of buried pulse (force)/sft), and the other is that of surface pulse, fsft). As published by Pekeris (Pekeris 1955), these two forces result in the completely different displacement fields at point x. In Fig. 7.2, examples of Lamb s solutions due to a buried step-function force, where/sO) = hsft), are given. The depth of the source, D, is 6 cm and the horizontal distance, R, is varied as 3 cm, 6 cm and 9 cm. Here P-wave velocity Vp is assumed as 4000 m/s and Poisson s ratio is 0.2. These material properties actually represent those of concrete. Near the epicenter, only P-wave and S-wave are observed as shown in Fig. 7.2 (a). 

Fig. 7.2. Lamb s solutions due to a buried step-function pulse.

Fig. 7,3. Lamb s solutions due to a surface step-function pulse. A solid curve shows the case v= 0.3 and a broken shows the case v= 0.25.

As discussed in Chapter 3, Breckenridge et al. [1981] developed a capacitance-type sensor of very flat response, by which they detected AE waves due to a break of glass capillary shown in Fig. 7.4. Later, the capillary break was replaced by the pencil-lead break by Hsu [1978]. As compared Fig. 7.3 with Fig. 7.4, first time, they showed that AE wave detected by the flat-type sensor due to the step-function force is actually identical to Lamb s solution due to the surface pulse. It was also demonstrated that Lamb s solution due to a buried pulse could be obtained by applying the force at the bottom of the block in Fig. 3.10. Thus, it is clarified by them that the displacement observed by the flat-type sensor due to capillary break or pencil-lead break is identical to G 33(x,yo,t). This implies that Green s function of the specimen can be empirically obtained by just applying the pencil-lead break and recording the displacements. 

The Fourier transform of the detected wave in Fig. 7.6 (a) is substituted into the left-hand of Eq. 7.19, and then the function S(f) is solved as U(f)/G(f), taking into account Lamb s solution G(f). Solution S(t) after the inverse Fourier transform of S(f) is shown in Fig. 7.12. In the figure, the solid curve is the function, S(t) = df(t)/dt, and the broken curve represents the function assumed in Eq. 7.14. Remarkable agreement is observed between the computed and the assumed. 

In the case that Eq. 8.2 is applied to the moment tensor analysis, the discrepancy between the half-space solution and the infinite-space solution should be studied. Accordingly, Lamb s solutions for a buried pulse are compared with Green s functions in an infinite space. 

As stated in Chapter 7, a code for computing Lamb s solution due to a buried pulse was already published (Ohtsu Ono 1984). An infinite-space solution is presented in the literature (Aki Richards 1980). Thus a solution UNij in an infinite space due to a step-function force H(t) is obtained as,... 

In the X1-X2-X3 coordinate system in Fig. 8.2, it assumed that step-function force f is applied in the X3-direction at the depth 6 cm, and elastic waves in the xs-direction are detected at three locations A, B, and C on the stress-free surface. The velocity of P-wave is 4000 m/s and Poisson s ratio is 0.2. Thus, to investigate the discrepancy between the far-field approximation and the solutions in a half space. Lamb s solution G33, infinite-space solution UN33, and the far-field solution UF33 are computed. 

Computed Green s functions at location A are given in Fig. 8.3. Lamb s solution due to a step-function force shows clear arrivals of P-wave and S-wave as similar to Fig. 7.2 (a). The amplitudes of the infinite-space solu-... 

Now the field due to a delta function source Qz = S(z — zo)5(x) is known as Lamb s problem, and at some distance from x = 0 the solution is known (Achenbach 1973). If the surface displacement of the surface wave generated by the concentrated load is wz x, C) and the depth of the crack is d, then for the distributed body forces, by superposition,... 

Here, the sphere center is instantaneously situated at point 0 the sphere center translates with velocity U, while it rotates with angular velocity (a r is measured relative to 0 its magnitude r is denoted by r. Moreover, f = r/r is a unit radial vector. The latter solution is derivable in a variety of ways e.g., from Lamb s (1932) general solution (Brenner, 1970). [Equation (2.12) represents a superposition (Brenner, 1958) of three physically distinct solutions, corresponding, respectively, to (i) translation of a sphere through a fluid at rest at infinity (ii) rotation of a sphere in a fluid at rest at infinity (iii) motion of a neutrally buoyant sphere suspended in a linear shear flow. The latter was first obtained by Einstein (1906, 1911 cf. Einstein, 1956) in connection with his classic calculation of the viscosity of a dilute suspension of spheres, which formed part of his 1905 Ph.D. thesis.]... 

Rate of permeation relative to tliat of maltose. Data adjusted to 100 s for maltose. The LamB-containing liposomes were added to buffer solutions containing 40 mM of die corresponding test sugars. 

Figure 8.12 shows the projected conversion of S02 to sulfate as a function of the volume of water per cubic meter of air available for conversion in the aqueous phase, covering a range typical of haze particles, fogs, and clouds for atmospheric lifetimes which are typical for each (Lamb et al., 1987). As expected from Eq. (M), the conversion increases with the water available in the atmosphere. As we shall see, the aqueous-phase oxidation does indeed predominate in the atmosphere under many circumstances. Equations (G) and (M) apply as long as the partial pressure of SOz in the gas phase, so,, is measured simultaneously with the solution concentration of S(IV). 

Trying to overcome the limitations of the Gaussian plume assumptions, Lamb presented a Green s function approach to the solution of the transport equation with both area sources at the ground and volumetric sources caused by reactions (17), Two other features included in the... 

If special precautions are taken to avoid contamination of the bubble surface, particle deposition by diffusion to the water surface is enhtinced by internal circulation. The internal flow can be calculated for very low bubble Reynolds numbers in the creeping flow approximation (Lamb, 1953). A solution ha.s been obtained to the equation of convective... 

Philippoff (87) applied the method of the reduced variables skillfully to show that smooth curves are obtained for G and G" if his result for polystyrene solutions in Aroclor and the result of Lamb and Matheson in toluene (82) are compared on a reduced scale. Although his reduction method includes c as well as 7 and t]s as variables, it is justified because... 

The confirmation by Lamb and Retherford of the inadequacy of the Dirac theory stimulated a re-examination of a theoretical problem to which only a very incomplete solution had so far been found the problem of the interaction between charged particles and the electromagnetic field. We shall briefly refer to the problem as it presented itself in classical physics, and then (following Weisskopf [135]) notice the further difficulties which the quantum theory introduces. Finally we shall see how these difficulties have been circumvented by the new quantum electrodynamics, and how a small correction is thereby introduced to the energy levels predicted by Dirac s theory. The new theory, however, is not a complete and logically satisfactory solution to the problems we shall state a difficulty of principle remains now, as formerly. 

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