investigated at a range of Mach numbers (1.5, 2, 2.5 and 3). Unless specified otherwise, the **shock** had a radius of 165 mm.
6.1 The First Disturbance
Figure 6.2 shows the typical behavior of a **cylindrical** **shock** with an initial radius of 165 mm on a circular wall. It is called typical because similar features were observed for all wall– **shock** ratios considered. In general, compression waves propagate up along the **shock** before the **shock** transitions to an MR thereafter it further transitions to a TRR corresponding to (Time < 20µs), (20µs < Time < 38µs) and (Time > 38µs) respectively in Figure 6.2 . Similar to **shock** **diffraction**, we can appeal to GSD and consider the first disturbance. As illustrated by Figure 6.3 and 6.4 , the first disturbance does not seem to be involved in the formation of the Mach **reflection** as was observed by Gruber and Skews with a plane **shock** on a **cylindrical** wall segment [ 53 ]. Where a **shock** reflects off of a sharp compressive corner, a reflected **shock** is immediately formed (in place of successive compression waves) thereby warranting the importance of the leading edge signal. It is this leading edge signal that the length scale criterion uses to determine when an MR forms and when it transitions into a TRR. When the **shock** encounters a curved convex wall with a zero initial wall angle the reflected **wave** is replaced by a series of compression waves. Consider Figure 6.5 , where the wall is composed of small **segments** (δs), each inclined to the previous by a small angle. Any curved wall can be approximated in this manner by choosing δs small enough. With this formulation there is not one, but infinitely many virtual corners along the wall each generating its own disturbances.

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Subsequently are listed some effects that cannot easily be explained by the simple **reflection** model. Figure 5 shows the SPH for both measuring paths i.e. Allouis–Kühlungsborn and Allouis–Juliusruh for a few years. Noticeable are the dif- ferences in springtime which are roughly 400 m in height. Taking into account that the **reflection** point of both paths is only 50 km apart these differences seems to be much too large. One would expect a few 10th of meters. Since it is an apparent height and **diffraction** is neglected. The involved real **reflection** area is probably greater than 50 km in diam- eter. If this is not the case, a considerable inclination of re- flecting layer must be assumed, which could be also a source for changes in field strength recorded at the receiver place. In this case one would record a value representing a tilting factor and not an apparent height.

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Abstract: In order to study the **propagation** law of explosive **shock** **wave** in underground turning roadway, the peak overpressure value in the roadway with specific turning angle was compared and analyzed by combining field monitoring experiment with ANSYS numerical simulation. The results show that the blast **wave** propagates forward in a stable plane **wave** before turning. Before the explosion air **shock** **wave** propagates to the turn, it follows the **propagation** law in the straight through roadway. After turning, the **diffraction** and **reflection** through the wall of the roadway will form a turbulence zone of 10-20m, and then continue to propagate forward in a stable plane **wave**. The turning roadway has a certain attenuation effect on the **propagation** of the **shock** **wave**. By analyzing the peak overpressure value before and after turning, the attenuation coefficient values of roadways at various turning angles are determined, That is, the attenuation coefficient values corresponding to the turning angles of 30°, 60°, 90°, 120° and 150° are 1.25, 1.31, 1.45, 1.50, 1.65, respectively, and the attenuation coefficient values are fitted with the roadway turning angle formula to obtain the quantitative calculation formula, which can provide reference for the safety of underground personnel, equipment and the design and production of mines.

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The stability of these converging shocks has been of much interest. Experiments by Perry and Kantrowitz (1951) showed that converging **cylindrical** **shock** waves are inherently unstable, and any disturbance along the **shock** front will result in a local increase in curvature, and even- tually cause the formation of reflected waves behind the original **shock** front. Schwendeman and Whitham (1987) determined both analytically and numerically that such a disturbance would form an MR, and showed that a converging **cylindrical** **shock** **wave** exposed to a series of regularly spaced disturbances would eventually result in a series of repeating polygons. This unstable behaviour was confirmed by experiments carried out by Watanabe and Takayama (1991). Figure 2.10 shows the resulting shape of a converging **cylindrical** **shock** **wave** some time after encountering four small, evenly-spaced disturbances. Schwendeman (2002) later showed that this behaviour also occurs in three dimensions, and that imploding spherical shocks exposed to regular disturbances would result in a series of repeating polyhedra.

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Consider the expansion **wave** formed at the upper **diffraction** edge (in the reference frame presented in the Figure) on the vertical plane of symmetry: if it were to meet the interior of the **shock** tube (a horizontal surface) at the lower side it would induce flow upward (toward the axis of the jet). Since this would violate the continuity of mass at that surface a compression fan or even a **shock** **wave** would have to form to induce the flow back parallel to the solid surface. If the expansion **wave** were to meet the inclined exit surface, the **shock** **wave** would form to induce the flow parallel to that surface. This description is based on an infinite-span, two-dimensional duct and thus, considering the fact that the **shock** tube is a **cylindrical** pipe, the expansion **wave** orientation and interaction will be much more complex here. Also, since the flow visualisation system can only visualise gradients in the plane normal to the optical axis, only the tangent edges of these secondary **shock** waves or compressions would be visible. Therefore, considering the Mach 1.47 incident **shock** **wave** results, part of the strong expansion formed at the upstream **diffraction** edge must meet the interior of the **shock** tube at the lower surface. As explained, a **shock** **wave** forms to turn the flow back parallel to the wall of the pipe. Careful observation of frame d) of this series bears this out since the apparent foot of the secondary **shock** **wave** is inside of the **shock** tube. The tangent edge of the secondary **shock** **wave** is curved toward the jet boundary by the interaction with the expansion fan and then meets the jet boundary where it seems to undergo weak **reflection** or simply termination in the turbulent flow there.

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A 1 µs xenon flash lamp was used as a light source and passed through a thin vertical slit, and then collimated (light rays made parallel) by parabolic mirrors. The collimated light was passed through the test section windows (and hence through the flow under investigation) and brought to a focus. A vertical knife-edge was placed at the focus point, and carefully positioned so some of the light is blocked. The light which passes the knife-edge is once again collimated with the use of a lens and is photographed directly with a camera. Vertical slits and knife-edges were used, since this configuration produces higher sensitivity, and therefore more defined images for vertical **shock** waves, which are predominant in the **reflection** under study. The flash lamp was triggered by the output signal of the time delay box.

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In Chapter 3 we demonstrated that steady-state longitudinal waves in an elastic layer and steady-state waves in layered materials are dispersive: the phase ve- locity depends on the frequency of the **wave**. Dispersive waves commonly occur in problems involving boundaries and interfaces. The Fourier integral theorem expresses a transient **wave** in terms of superimposed steady-state components with a spectrum of frequencies. If there is dispersion, each component prop- agates with the phase velocity corresponding to its frequency. As a result, a transient **wave** tends to spread, or disperse, as it propagates. This is the origin of the term dispersion. Because the diﬀerent frequency components propagate with diﬀerent velocities, it is not generally possible to deﬁne a velocity of a transient **wave**. However, we will show that a transient **wave** having a narrow frequency spectrum propagates with a velocity called the group velocity.

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Eurasian National University, 5 Munaitpassov St., Astana, Kazakhstan Abstract—The classical problem for **diffraction** of a plane **wave** with an arbitrarily oriented **wave** vector at a strip is considered asymptotically by Wiener-Hopf method. The boundary-value problem has been broken down into distinct Dirichlet and Neumann problems. Each of these boundary-value problems is consecutively solved by a reduction to a system of singular boundary integral equations and then to a system of Fredholm integral ones of second kind. They are solved effectively by a reduction to a system of linear algebraic equations with the help of the etalon integral and the saddle point method.

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Electromagnetic **wave** **propagation** is first analyzed in a composite material mde of chiral nano-inclusions embedded in a dielectric, with the help of Maxwell-Garnett formula for permittivity and permeability and its reciprocal for chirality. Then, this composite material appears as an homo-geneous isotropic chiral medium which may be described by the Post constitutive relations. We analyze the **propagation** of an harmonic plane **wave** in such a medium and we show that two different modes can propagate. We also discuss harmonic plane **wave** scattering on a semi-infinite chiral composite medium. Then, still in the frame of Maxwell-Garnett theory, the **propagation** of TE and TM fields is investigated in a periodic material made of nano dots immersed in a dielectric. The periodic fields are solutions of a Mathieu equation and such a material behaves as a **diffraction** grating.

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A. Sasoh, K. Takayama and T. Saito [12] have conducted experiments and numerical simulation concerning the curvature of the diffracted **shock** for a **shock** Mach number equal to 1.15 and angle of bend =15 0 (They have used in place of ). They have also presented results connecting triple point angle χ and the angle both experimentally and theoretically.

approximately 1500 times per mile during running (McKeag & Dolan, 1989), or 930 times per kilometre. Each impact subjects the individual to forces of up to 2.8 times their body weight (Cavanagh & Lafortune, 1980); these forces are subsequently transmitted proximally through the tissues of the body as a stress **wave**, commonly referred to as a '**shock** **wave**' (Derrick et al., 1998; Dufek et al., 2009; Flynn et al., 2004; Hamill et al., 1995; Holmes & Andrews, 2006; Lafortune, 1991; Lafortune et al., 1995, 1996a, b; McMahon et al., 1987; Mercer et al., 2002, 2003a, b; Milner et al., 2006; Verbitsky et al., 1998; Voloshin et al., 1998; Voloshin & Wosk, 1982; Whittle, 1999; Wosk & Voloshin, 1981). As the **shock** **wave** propagates through the body, the rigid and soft tissues act to attenuate the signal (Dufek et al., 2009; Hamill et al., 1995). Previous studies using animals have indicated that repetitive loading may lead to cartilage and joint degeneration (Radin et al., 1973; Serink et al., 1977) and bone microdamage (Burr et al., 1985). The results of studies using human participants also imply a relationship between injury and impact forces and the associated **shock** **wave**, in that runners with a stress fracture history experience greater impact forces (Grimston et al., 1991), loading rates, and tibial **shock** (Milner et al., 2006).

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To investigate the effect of 3-D relief on a sharp fin SBLI, a myriad of experimental techniques are employed to analyze both the surface and off-body mean flows of a fin placed on a **cylindrical** surface. Oil flow streaklines provide a qualitative picture of the shear contours and separation entities, while pressure sensitive paint (PSP) provides a quantitative picture of the surface pressure field. Moreover, planar laser scattering (PLS) and particle image velocimetry (PIV) are used to determine how the off-body flow corresponds to the patterns seen on the surface. A technique was developed to calibrate the PLS intensity fields to density fields to enhance its quantitative value, thus providing deeper insight into the physics at play. When possible, all processing is done with in-house codes developed in MATLAB; the notable exception being processing of the velocity fields, which was done in DaVis 8.4. Because extreme maneuvers of vehicles would represent a wide range of **shock** strengths, separation regimes, and distortion magnitudes, extensive work is presented in this manuscript in which the Mach number, fin angle, extent of 3-D relief, and perturbation size are varied. Complementary planar fin SBLI and computational RANS simulations are presented as well as a basis of comparison to the fin-on-cylinder configuration and to extend the results into regions unobtainable in the experiments. A consolidated list of all of the experimental and computational methods is provided in chapter 2.

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M oderately strong shocks can be produced by either electric or magnetic driving, or, as usually le the case, by a combination of... electrical conductivity, the heatin[r]

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An electromagnetic wave signal may be represented by a function of space and time.. u(x,t) where u might be one component of the electric field, for example..[r]

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A linear trend of phase shift with frequency will lead to a phase speed that does not vary with frequency, but only if the phase shift tends to zero as the frequency tends to zero. The intercept on the phase shift axis is clearly nonzero in Figure 5.3. The nonzero phase shift at zero frequency is likely due to the method in which the phase shift is calculated. The pressure signal at the right wall is compared to the displacement of the left wall (as opposed to comparing the pressures at each wall) using a cross-correlation. This comparison was used because of the weak signal amplitude at the left wall due to resonance effects (see Figure 5.2 and Section 5.3) which led to poor results in the cross-correlation. Comparing dissimilar measurements (displacement vs. essentially acceleration) may introduce an additional phase. This additional phase was calculated from a linear fit of the phase vs. frequency curve and subtracted from the phase shift before determining the phase speed. The phase speed calculated in this manner is shown in Figure 5.4. As expected, the speed is, on average, constant with frequency and equal to the values of the group velocity found above. The roughly constant phase speed corresponds to nondispersive **propagation** as discussed in preceding chapters. The deviations about the linear trend in the phase versus frequency curves are mirrored in this plot as variations around this constant speed.

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• Multipath is a term used to describe the multiple paths a radio **wave** may follow between transmitter and receiver. Such **propagation** paths include the ground **wave**, ionospheric refraction, reradiation by the ionospheric layers, **reflection** from the Earth's surface or from more than one ionospheric layer, etc.

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Vallencien 4 has reported on difficulty in visualising stones in 10% and inability in 3% of patients, using ultrasound monitoring. He also noted that the rib interfered with localisation in7% of patients. Third generation lithotriptor have more powerful ultrasounds and the ability to move the in line scanner obliquely and radially to overcome these problems. All stones larger than 3mm can be easily visualised. Radiological visualisation has an edge over ultrasound in that ureteric stones are easily visualised and treated. Graff et al 8 treat non obstructing ureteral stones in situ with a 13% second treatment rate and 5.9% ancillary procedure rate. In situ fragmentation of ureteral stones, however, gives a 60-90% successful fragmentation rate as against 98-99% successful fragmentation of renal calculi 9,10 . Stones impacted in the ureter are more resistant to **shock** **wave** fragmentation 11 . This is because of lack of fluid filled space surrounding the stone which therefore has little room to expand 11 . Urine does not permeate the stone and the thick muscle near the ureter absorbs sound. Non opaque stones can be identified by continuous irrigation of the pelvis with dye introduced through a ureteric catheter.

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In [11], Gregory’s method is described as “rather cumbersome” and in [9] it is noted that Gregory’s solution “contains complicated contour integrals which are diﬃcult to evaluate numerically”. The latter statement is certainly no longer true, though care does need to be taken, and while it is true that some lengthy calculations are required, Gregory’s method seems no more cumbersome than any of the other techniques available. In this paper we have extended Gregory’s approach to the case of oblique incidence. This introduces some additional complications, most notably that the field must now be expressed in terms of three coupled potentials. Also, the nature of the far field changes depending on the angle of incidence. For normal and small angles of incidence the scattered field is made up of reflected and transmitted Rayleigh waves together with **cylindrical** P- and S-waves. Above some critical angle the P-**wave** disappears and then above a second critical angle, the S-**wave** is not present either. Our key objective is to illustrate this phenomenon by showing how the scattered energy is distributed between these diﬀerent waves as the angle of incidence or the frequency varies. It is possible to derive relations which express conservation of energy for our problem and in order to do this we have extended results of [14] which were derived for normal incidence in a manner analogous to Newman’s work on water waves [15]. An alternative approach to the derivation of these relations, again for normal incidence, is given in [16].

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