Mach Wave Fronts Research Articles

Generalizing the Mach cone for sources traveling along arbitrary trajectories

The concept of Mach cone arises when a wave source travels along a linear trajectory through a given medium with speed larger than the wave speed in that medium, and a shock wave is created. It is commonly associated to mechanical waves (specially sound), although also appears in electromagnetic waves as in the case of Cerenkov radiation. In all cases, the conical shape (or arrow tip shape in 2D media) is a consequence of the linear trajectory of the source. Here, we generalize this situation and determine the shape of the Mach wave front for sources traveling along arbitrary trajectories. To do so, instead of waves, we consider that the source emits rays as in geometrical optics, and determine the equations describing the points in the space that are reached simultaneously by the rays, which form the Mach wave front. To show the applicability of our general results, we obtain explicitly the Mach wave fronts for three different trajectories: linear, circular and sinusoidal.

Offshore impact pile driving as a source of opportunity for geoacoustic investigations

Noise generated by offshore impact pile driving can radiate into and propagate through the air, water, and sediment. Most of the recent studies have focused on predicting acoustic pressure field in water to assess environmental impacts. In this study, we focus on the propagation of acoustic energy along the interface between water and ocean bottom. We modeled the interface wave propagation using the commercial Finite Element (FE) code, Abaqus 6.11. A field test is planned for this summer using a scaled model impact pile driving off the dock in the Bay Campus University of Rhode Island (URI). Interface data (particle velocities at the water-sediment interface) will be collected using the Shear Measurement System. Our efforts will focus on identifying the arrivals corresponding to various wave types using data-model comparison. In addition to the interface waves, we will also model the Mach wave front arrival angle. We will explore the possibility of utilizing the arrival times and angles corresponding to these wave types for setting up an inverse problem. This test will also verify the possibility calculating compressional wave speed using a single three-axis geophone by measuring arrival angle of Mach wave front at the interface generated by impact pile driving. [Work sponsored by the Link Foundation Ocean Engineering and Instrumentation PhD Fellowship program.]

Photo-acoustic study of subshear and supershear ruptures in the laboratory

We report supershear ruptures observed in spontaneously nucleating laboratory microearthquakes and describe the signature of the associated Mach wavefront radiation. Transducers detect the wavefield both close and at a distance from the fault. The rupture velocities are inferred from either photoelastic high-speed imaging or using the acoustic recordings; both methods yield compatible estimates of sufficient accuracy to discriminate between sub-shear and supershear ruptures. The acoustic records allow to characterize the Mach wavefront radiated from the supershear rupture front, in particular its amplitude and decay. Velocity functions recovered by integrating high frequency accelerometer signals in the case of supershear ruptures, consist in a double-pulse function: a first pulse traveling at supershear velocity followed by a second pulse traveling at the Rayleigh wave velocity. Conversely, the sub-shear event is characterized by a single pulse. Finally, we perform numerical simulations of our experiment using a prescribed supershear rupture velocity. The synthetic waveforms obtained from these simulations, including the Mach wave amplitude and phase, yield a satisfactory fit to the experimental results.

Coherence of Mach fronts during heterogeneous supershear earthquake rupture propagation: Simulations and comparison with observations

We study how heterogeneous rupture propagation affects the coherence of shear and Rayleigh Mach wavefronts radiated by supershear earthquakes. We address this question using numerical simulations of ruptures on a planar, vertical strike‐slip fault embedded in a three‐dimensional, homogeneous, linear elastic half‐space. Ruptures propagate spontaneously in accordance with a linear slip‐weakening friction law through both homogeneous and heterogeneous initial shear stress fields. In the 3‐D homogeneous case, rupture fronts are curved owing to interactions with the free surface and the finite fault width; however, this curvature does not greatly diminish the coherence of Mach fronts relative to cases in which the rupture front is constrained to be straight, as studied by Dunham and Bhat (2008a). Introducing heterogeneity in the initial shear stress distribution causes ruptures to propagate at speeds that locally fluctuate above and below the shear wave speed. Calculations of the Fourier amplitude spectra (FAS) of ground velocity time histories corroborate the kinematic results of Bizzarri and Spudich (2008a): (1) The ground motion of a supershear rupture is richer in high frequency with respect to a subshear one. (2) When a Mach pulse is present, its high frequency content overwhelms that arising from stress heterogeneity. Present numerical experiments indicate that a Mach pulse causes approximately anω−1.7high frequency falloff in the FAS of ground displacement. Moreover, within the context of the employed representation of heterogeneities and over the range of parameter space that is accessible with current computational resources, our simulations suggest that while heterogeneities reduce peak ground velocity and diminish the coherence of the Mach fronts, ground motion at stations experiencing Mach pulses should be richer in high frequencies compared to stations without Mach pulses. In contrast to the foregoing theoretical results, we find no average elevation of 5%‐damped absolute response spectral accelerations (SA) in the period band 0.05–0.4 s observed at stations that presumably experienced Mach pulses during the 1979 Imperial Valley, 1999 Kocaeli, and 2002 Denali Fault earthquakes compared to SA observed at non‐Mach pulse stations in the same earthquakes. A 20% amplification of short period SA is seen only at a few of the Imperial Valley stations closest to the fault. This lack of elevated SA suggests that either Mach pulses in real earthquakes are even more incoherent that in our simulations or that Mach pulses are vulnerable to attenuation through nonlinear soil response. In any case, this result might imply that current engineering models of high frequency earthquake ground motions do not need to be modified by more than 20% close to the fault to account for Mach pulses, provided that the existing data are adequately representative of ground motions from supershear earthquakes.

Can dislocations accelerate through the shear-wave speed “barrier”?

The question of whether a dislocation can accelerate through the shear-wave speed “barrier” is addressed by analyzing the transient motion of a Volterra dislocation at the instant when the velocity equals the shear-wave speed in the presence of acceleration, which requires an asymptotic analysis at a double root (–at the transition from subsonic to supersonic a pair of complex conjugate roots becomes a double real —). The stresses carried by the forming Mach wave fronts depend on the acceleration at this instant and are found to be O( ln \(r /r^{1/2}\)) singular for a Volterra dislocation both screw and edge. The energy required to push the dislocation through the shear-wave speed “barrier” is determined by means of the “contour-independent” dynamic J integral which defines the self-force on a moving defect, and is obtained as a function of the acceleration as it crosses the “barrier”. While for a Volterra dislocation the energy-rate is singular at this instant, for a more physically realistic ramp-core “smeared” dislocation model, approximating the Volterra dislocation by a delta sequence, this energy rate is obtained by convolution and is finite, with the same result obtained by the theory of distributions. Thus, crossing the “barrier” is theoretically possible as recent experimental evidence in the literature suggests. A “cut-off” constant that remains undetermined will be found in a multiscale analysis by the matching of the self-force based on atomistic calculations modeling the core to the continuum far-field one obtained here. For decelerating motion through the shear-wave speed “barrier” this energy is released as dissipation.

The energetics of dislocations accelerating and decelerating through the shear-wave speed barrier

The question of whether a dislocation can accelerate through the shear-wave speed “barrier” is addressed by analyzing the transient motion at the instant when the velocity equals the shear-wave speed in the presence of acceleration. The stresses carried by the forming Mach wave fronts depend on the acceleration at this instant, and the energy required to push the dislocation through the shear-wave speed barrier is determined by means of the “contour-independent” dynamic J integral, which defines the self-force on a moving defect, and is obtained as a function of the acceleration as it crosses the barrier. For decelerating motion through the shear-wave speed barrier this energy is released as dissipation.

Shear Mach wave characterization for kinematic fault rupture models with constant supershear rupture velocity

SUMMARY We present the specific amplitude and waveform characteristics of near-source shear Mach waves generated by a kinematic model of supershear rupture at constant velocity ν. Asymptotic analytical solutions are provided for the Mach wave amplitudes, in relationship to the geometrical singularities carried by the S-wave isochrones on the fault plane. The solution is dominated by waves radiated near a critical point source A defined by cos(θ) =β/ν, where θ is the angle between the rupture ray and the S-wave ray normal to the rupture front at A, and β is the S-wave velocity (the Mach angle). The far-field, dominant velocity field of the Mach wave related to the mode II component of the slip is proportional to the slip velocity at A and to cos(2θ)/sin(θ). Thus, the related wave front carries the motion on the fault plane at large distances, with little attenuation, within a beam of Mach wave rays characterized by their angle θ. Numerical calculation of the complete field has been achieved up to 4 Hz, in a homogeneous elastic half-space, and for a vertical strike-slip fault 50 km long equivalent to a magnitude of 7.1. It confirms these theoretical developments, and shows that the peak acceleration and velocities are at least twice that of a standard sub-Rayleigh rupture at 10 km and up to 5 times its value at 30 km. Although the diffusion and diffraction of S waves in the real crust are expected to reduce the coherence of the Mach wave front and hence its peak amplitude, especially at large distances and for high frequencies, our analytical and numerical developments demonstrate that supershear rupture can produce unusually large levels of ground motion at distances ranging from 10 km to a few tens of kilometres, within the Mach wave beam.

Mach wave radiation of nonlinearly evolving supersonic instability modes in shear layers

Shear flows at sufficiently high Mach numbers support instability waves which travel supersonically relative to the ambient free stream. Such supersonic modes are known to produce intense far-field sound in the form of Mach wave radiation. In this paper, the nonlinear evolution of supersonic modes and the associated Mach wave radiation are analysed in a self-consistent fashion by using the high-Reynolds-number matched asymptotic expansion in conjunction with the multiple-scale method. Attention is focused on the relatively weak disturbances for which the nonlinear effect is comparable with the non-parallel-flow effect. The nonlinear spatial development of the mode is described by an amplitude equation in which the nonlinear term is contributed by the critical layer. The directivity of the radiated Mach waves in the far field is determined explicitly in terms of the amplitude function. The analysis is applicable to plane mixing layers, boundary layers, and planar and circular jets. In particular, it is shown that for the last two flows, the radiated Mach waves are most intensive in a beam which is perpendicular to the Mach wave front and emanates from the streamwise position at which the instability mode attains its maximum amplitude. The theoretical results are compared with direct numerical simulation and experimental data, and favourable qualitative and quantitative agreement is obtained.

A non-uniformly moving line force in an anisotropic elastic solid

The transient motion due to a line load moving through an unbounded anisotropic elastic body is considered. The force is applied suddenly in the body, and then moves off with non-uniform speed. Explicit expressions for the displacements are derived using the reciprocal theorem. A parametric form of the Mach wavefronts is presented when the force moves in a fixed direction, and first-motion responses are obtained for different wave arrivals by asymptotic methods.

The Transient Motion of a Ramp-Core Supersonic Dislocation

The transient motion of a ramp-core dislocation spreading from −∞<x<∞ and jumping from rest to a supersonic speed in the x direction, is investigated by analysis on the complex transform plane. The new result of this analysis is that, instead of the Mach cone of the Volterra dislocation, there are two lines of discontinuity (of the stress) propagating in the ±z directions, inside of which there are arctan delta sequence (ε≠0) radiated supersonic fields, as well as subsonic fields. The lines of discontinuity arise at the tangent point of the Mach wavefront (ε=0) to the cylinder with radius r=c2t.

An ultrasonic caliper for borehole measurement

We report supershear ruptures observed in spontaneously nucleating laboratory microearthquakes and describe the signature of the associated Mach wavefront radiation. Transducers detect the wavefield both close and at a distance from the fault. The rupture velocities are inferred from either photoelastic high-speed imaging or using the acoustic recordings; both methods yield compatible estimates of sufficient accuracy to discriminate between sub-shear and supershear ruptures. The acoustic records allow to characterize the Mach wavefront radiated from the supershear rupture front, in particular its amplitude and decay. Velocity functions recovered by integrating high frequency accelerometer signals in the case of supershear ruptures, consist in a double-pulse function: a first pulse traveling at supershear velocity followed by a second pulse traveling at the Rayleigh wave velocity. Conversely, the sub-shear event is characterized by a single pulse. Finally, we perform numerical simulations of our experiment using a prescribed supershear rupture velocity. The synthetic waveforms obtained from these simulations, including the Mach wave amplitude and phase, yield a satisfactory fit to the experimental results.

Estimating pressure in nonsteady reflection of a shock wave

A comparative analysis of the results of physical [1–4] and mathematical experiments [5–8] is used to elucidate the mechanism of additional pressure lift at a Mach wavefront. The possibility and range of application of simplified flow models for the estimation of the pressure at certain characteristic points of the reflection surface (for example, a prism, a cylinder, or a sphere) was investigated.

Irregular interaction of weak shock waves of different intensities: PMM vol. 38, n≗ 1, 1974, pp. 105–114

The problem of irregular interaction of weak shock waves, which occurs in the analysis of interpenetration of two waves of different intensities at small interaction angle [1, 2], is considered. It is not possible to solve this problem in linear configuration when the region adjacent to the Mach wave front shrinks to a point, which results in it becoming a nonlinear problem. Behavior of the solution throughout the interaction region is analyzed by the method of matching asymptotic expansions [3, 4]. The external problem is solved in linear formulation. A boundary value problem for the system of nonlinear equations of short waves [5], which takes into account the linking of its solution with the linear external problem and with solutions in the neighborhood of reflected fronts at the inner region boundary, is formulated for the inner region in the neighborhood of the Mach wave front. The effect of the initial state parameters on the pattern of flow is investigated and an approximate solution of the problem is derived.

Investigation of Supersonic Jet Noise Using Optical Holography

Holographic techniques have been used to obtain three-dimensional visualizations of the noise field generated by a cold supersonic turbulent jet. Double exposure holographic interferograms of an underexpanded jet display conical fringes emanating from the jet, which are believed to be Mach wavefronts. These fronts appear to originate at disturbances on the core of the jet. Measurements of Mach angles indicate that the predominant convection velocity of the disturbances is approximately 80% of the jet velocity.