Inhomogeneous Wave Field Research Articles

Array processing for wave vectors and modes in structural acoustics

Array processing methods have emphasized beamforming and directional spectrum estimation for equally spaced, linear arrays with single component sensors that are in a homogeneous signal field with long duration data. Structural acoustics problems often have just the opposite characteristics with irregular spacing of multicomponent sensors that are in inhomogeneous wave fields with transient data. These eliminate many array processing methods and require generalization of others. Recently, array processing methods such as matched field and matched mode, slowness‐time spectra, and singular value decomposition have been advanced and used for structural acoustics. This presentation will give an overview of advanced array processing methods with an emphasis on those applicable to problems in structural acoustics. [Work supported by ONR.]

ON THE USE OF VOLUMETRIC STRAIN METERS TO INFER ADDITIONAL CHARACTERISTICS OF SHORT-PERIOD SEISMIC RADIATION

Volumetric strain meters (Sacks-Evertson design) are installed at 15 sites along the San Andreas fault system, to monitor long-term strain changes for earthquake prediction. Deployment of portable broadband, high-resolution digital recorders (GEOS) at several of the sites extends the detection band for volumetric strain to periods shorter than 5 x 10 -2 sec and permits the simultaneous observation of seismic radiation fields using conventional short-period pendulum seismometers. Simultaneous observations establish that the strain detection bandwidth extends from periods greater than 107 seconds to periods near 5 x 10 -2 sec with a dynamic range exceeding 140 dB. Measurements of earth-strain noise for the period band, 107 to 10 -2 sec, show that ground noise, not instrument noise, currently limits the measurement of strain over a bandwidth of more than eight orders of magnitude in period. Comparison of the short-period portion of earth-strain, noise spectra (20 to 5 x 10 -2 sec) with average spectra determined from pendulum seismometers, suggest that observed noise is predominantly dilatational energy. Recordings of local and regional earthquakes indicate that dilatometers respond to P energy but not direct shear energy and that straingrams can be used to resolve superimposed reflected P and S waves for inference of wave characteristics not permitted by either sensor alone. Simultaneous measurements of incident P- and S-wave amplitudes are used to introduce a technique for single-station estimates of wave field inhomogeneity, free-surface reflection coefficients and local material P velocity. Estimates of these parameters derived for the North Palm Springs earthquake (Mw 5.9) respectively for an incident P wave of 29 ° are -85 °, 1.71, 2.9 km/sec, and for an incident S wave of 17 ° are 79 °, 0.85, 2.9 km/sec. The empirical estimates of reflection coefficients are consistent with model estimates derived using an anelastic half-space model with incident inhomogeneous wave fields.

An exact anelastic model for the free-surface relfection ofPandS-I waves

AbstractExact anelastic solutions incorporating inhomogeneous waves are used to model numerically S-I and P waves incident on the free surface of a low-loss anelastic half-space. Anelastic free-surface reflection coefficients are computed for the volumetric strain and displacement components of inhomogeneous wave fields. For the problem of an incident homogeneous S-I wave in Pierre shale, the largest strain and displacement amplitudes for the reflected P wave occur at angles of incidence for which the particle motion for the reflected inhomogeneous P wave is elliptical (minor/major axis = 0.6), the specific absorption (QP−1) is greater (300 per cent) and the velocity is less (25 per cent) than those for a corresponding homogeneous P wave, the direction of phase propagation is not parallel to the free surface, and the amplitude of the wave shows a significant increase with depth (6 per cent in one wavelength). Energy reflection coefficients computed for this low-loss anelastic model show that energy flow due to interaction of the incident and reflected waves reach maxima (30 per cent of the incident energy) near large but nongrazing angles of incidence. For the problem of an incident homogeneous P wave in Pierre shale, the inhomogeneity of the reflected S wave is shown not to contribute to significant variations in wave field characteristics over those that would be expected for a homogeneous wave.

Influence of welded boundaries in anelastic media on energy flow, and characteristics of P, S‐I, and S‐II waves: Observational evidence for inhomogeneous body waves in low‐loss solids

A general computer code, developed to calculate anelastic reflection‐refraction coefficients, energy flow, and the physical characteristics for general P, S‐I, and S‐II waves, quantitatively describes physical characteristics for wave fields in anelastic media that do not exist in elastic media. Consideration of wave fields incident on boundaries between anelastic media shows that scattered wave fields experience reductions in phase and energy speeds, increases in maximum attenuation and Q−1, and directions of maximum energy flow distinct from phase propagation. Each of these changes in physical characteristics are shown to vary with angle of incidence. Finite relaxation times for anelastic media result in energy flow due to interaction of superimposed radiation fields and contribute to energy flow across anelastic boundaries for all angles of incidence. Agreement of theoretical and numerical results with laboratory measurements argues for the validity of the theoretical and numerical formulations incorporating inhomogeneous wave fields. The agreement attests to the applicability of the model and helps confirm the existence of inhomogeneous body waves and their associated set of distinct physical characteristics in the earth. The existence of such body waves in layered, low‐loss anelastic solids implies the need to reformulate some seismological models of the earth. The exact anelastic formulation for a liquid‐solid interface with no low‐loss approximations predicts the existence of a range of angles of incidence or an anelastic Rayleigh window, through which significant amounts of energy are transmitted across the boundary. The window accounts for the discrepancy apparent between measured reflection data presented in early textbooks and predictions based on classical elasticity theory. Characteristics of the anelastic Rayleigh window are expected to be evident in certain sets of wide‐angle, ocean‐bottom reflection data and to be useful in estimating Q−1 for some ocean bottom reflectors.

GeneralP, type-IS, and type-IISwaves in anelastic solids; inhomogeneous wave fields in low-loss solids

AbstractThe physical characteristics for general plane-wave radiation fields in an arbitrary linear viscoelastic solid are derived. Expressions for the characteristics of inhomogeneous wave fields, derived in terms of those for homogeneous fields, are utilized to specify the characteristics and a set of reference curves for general P and S wave fields in arbitrary viscoelastic solids as a function of wave inhomogeneity and intrinsic material absorption. The expressions show that an increase in inhomogeneity of the wave fields causes the velocity to decrease, the fractional-energy loss (Q−1) to increase, the deviation of maximum energy flow with respect to phase propagation to increase, and the elliptical particle motions for P and type-I S waves to approach circularity. Q−1 for inhomogeneous type-I S waves is shown to be greater than that for type-II S waves, with the deviation first increasing then decreasing with inhomogeneity. The mean energy densities (kinetic, potential, and total), the mean rate of energy dissipation, the mean energy flux, and Q−1 for inhomogeneous waves are shown to be greater than corresponding characteristics for homogeneous waves, with the deviations increasing as the inhomogeneity is increased for waves of fixed maximum displacement amplitude. For inhomogeneous wave fields in low-loss solids, only the tilt of the particle motion ellipse for P and type-I S waves is independent to first order of the degree of inhomogeneity. Quantitative estimates for the characteristics of inhomogeneous plane body waves in layered low-loss solids are derived and guidelines established for estimating the effect of inhomogeneity on seismic body waves and a Rayleigh-type surface wave in low-loss media.

Asymptotic theory for inhomogeneous waves

The conventional asymptotic theory for propagation of high-frequency fields is based on a local description in terms of homogeneous plane waves A(r) exp [ik_{0}S(r)] , where k_{0} is the (large) free space wavenumber, A(r) is a spatially dependent amplitude, and the phase S(r) is real. The conventional theory does not accommodate the more general class of fields that behave locally like inhomogeneous plane waves with complex phase S(r)= R(r) + iI(r) , where R determines the propagation of the equiphase surfaces and I describes the attenuation. This paper develops an asymptotic theory for inhomogeneous wave fields in lossless media, to be termed evanescent fields. Such fields are encountered, for example, in connection with Gaussian beams and with phenomena on the exterior of surface wave structures or on the dark side of caustics. The scalar wave equation is used to derive eikonal and transport equations for S and A , respectively, and it is shown how the latter equations may be integrated with the aid of trajectories tangent to the direction of \nablaR , which differs slightly from that for the local power flow. Detailed application of the theory is made to two-dimensional scattering of a weakly evanescent incident plane wave by a curved boundary in a homogeneous medium. The phase propagation paths for the reflected field are determined explicitly and are found to possess curvature and points of inflection; these characteristics are shown to be predictable from basic attributes of evanescent wave propagation. For the special case of a circular cylinder, the subsequently constructed reflected field is found to agree with the asymptotic expansion of the rigorous solution, thereby confirming the validity of the theory for weakly evanescent fields. The rigorous solution, valid for arbitrary evanescent decay and obtained from known results for ordinary plane wave scattering by analytic continuation of the incidence angle to complex values, reveals that both the reflected and creeping wave fields should be viewed in a restrictive manner when the evanescent decay is large. However, for weak evanescent decay, these fields retain their customary significance and permit their construction by local evanescent field tracking. It is interesting to observe that in contrast to the nonevanescent case, the creeping waves provide a field contribution exceeding that of the incident or reflected waves in certain portions of the illuminated region.