Effective Wave Speed Research Articles

Lower solar chromosphere-corona transition region. II - Wave pressure effects for a specific form of the heating function

Lower transition region models with a balance between mechanical heating and radiative losses are expanded to include wave pressure effects. The models are used to study the simple damping length form of the heating function. The results are compared to the results obtained by Woods et al. (1990) for solutions in the lower transition region. The results suggest that a mixture of fast-mode and slow-mode waves may provide the appropriate heating mechanism in the lower transition region, with the decline in effective vertical wave speed caused by the refraction and eventual total reflection of the fast-mode wave resulting from the decreasing atmospheric density.

Wave Propagation Through Elastically-Anisotropic Fluid-Saturated Porous Rocks

An elastically-anisotropic sedimentary rock is modeled by a simple cubic packing of identical, contacting spherical particles. The connected pore space is filled with an inviscid, compressible fluid. A set of averaged equations is derived to relate the constitutive and dynamic coupling coefficients, and hence also the effective wave speeds in any given direction explicitly to the microstructural properties of the rock considered. Simple, explicit results are obtained when the propagation of either a purely longitudinal or a purely transverse wave is considered.

The construction of dynamic effective medium theories (EMT) for inhomogeneous fluid or elastic continua

A deterministic technique is described that predicts the dynamic (i.e., frequency‐dependent) effective properties of inhomogeneous media. Situations are considered in which a host medium sustaining a distribution of spherical inhomogeneities is either fluid or solid. In the Rayleigh region (viz., a/λ < 1) in which most effective medium theories (EMTs) hold, inhomogeneities of any shape can be assumed spherical without much error. However, assuming them to be penetrable implies that they realistically admit interior fields, which are coupled to the exterior ones by stress and displacement boundary conditions. The outcome of dynamic EMTs is a set of expressions to predict the frequency dependence of the effective (sound) wave speed c̄eff, and the effective (sound) wave attenuation ᾱeff in the mixture. These quantities emerge from the real and imaginary parts of the effective wavenumber k̄eff, which are connected by Kramers‐Kronig type relations, as has been discussed by Dr. Beltzer. An important consideration for the practical usefulness of an EMT is its simplicity. Often, the resulting predictive expressions are so cumbersome that they can only be evaluated by means of quite formidable computer codes. The principal EMT to the discussed here, yields closed‐form simple predictive expressions for c̄eff and ᾱeff, which, in many instances, are valid over significantly broad frequency bands. These expressions reduce to (and contain) almost all existing earlier results for static (f→0) cases. The (present form of this) EMT is constructed by exploiting the presence of the monopole resonance of the inhomogeneities, and thus it works best in the cases in which this resonance is dominant. This EMT has to be further refined to account for cases where the dipole resonance is dominant. In this regard, Dr. Martinez, in a later lecture, will discuss an approach to more accurately deal with heavy/rigid inclusions in a rubberlike matrix. [Work supported by NSWC and ONR.]

Resonance scattering of elastic waves by an elastic inclusion in an elastic matrix. Numerical calculations

The scattering of an incident plane ultrasonic (longitudinal) wave by an elastic spherical inhomogeneity contained within an elastic matrix is studied. The emphasis is on the computation and analysis of basic multiple cases that result when different material behaviours are present in the matrix and in the inclusion. These calculations are useful in underwater acoustic applications. The behaviors are dominated by the soft or rigid backgrounds of the resonance scattering theory (RST). The first three multiple coefficients appearing in the expansions for the total elastodynamic fields developed around the inhomogeneity during the scattering process have been calculated in suitable frequency bands in all the cases considered. The examination of modulus, the real parts, and the imaginary parts of these (complex) coefficients under the RST approach allows the quantitative assessment of the conditions under which monopole or dipole resonances will occur and their relative magnitudes. The decomposition of the multiple coefficients into their resonance and background portions shows that it is the upward frequency shift of the background curves that controls the dominance of either radial (monopole) or translation (dipole) oscillations of the inclusion. This has an effect on the dispersion curves of the composite, which develop optical as well as acoustical branches. The real and imaginary parts of the multiple coefficients are respectively proportional to the attenuation and the effective wave speed in this simple inhomogeneous composite.

Wave Propagation Through Fluid Saturated Porous Rocks

A sedimentary rock is modeled by a random packing of identical spherical particles. The connected pore space is filled with an inviscid, compressible fluid. A low-frequency expansion technique is used to calculate the effective wave speeds explicitly in terms of the microstructural properties of the rock considered. The effect of both the pore fluid and the initial confining pressure to which the rock is subjected can be included in the calculations.

Compressible and Leaky Trains in Tunnels

It is hypothesized that phase differences commonly observed between computed and measured pressure histories in railway tunnels may be caused in part by reduced acoustic wave speeds in the annular regions of flow alongside trains. Such reductions in wave speed are shown to be a possible consequence of the compressibility of trains and/or the leakage of air into and out of trains. These effects are shown to be capable in principle of reducing the effective wave speed by a factor of (1-β)1/2 where β denotes the train-tunnel area blockage ratio. The potential influence of reduced wave speeds due to these causes is shown to be much greater in short tunnels than in long tunnels. An alternative explanation for the discrepancies observed in long tunnels has yet to be found.

An evaluation of matched field processor performance with environmental mismatch

The matched field processor is enjoying a technical renaissance because of increased computer computation speeds and environmental modeling capability. Its performance, however, is degraded by error or mismatch in waveguide steering vectors (theoretical fields) and by ambient noise. The effects of error in sound velocity profile, water depth, array depth, and sensor positions will be discussed in relation to their effect on source location bias, peak height, and peak width as a function of mismatch strength, frequency, range, and array length. Noise effects are not considered. To gain insight, the processor output was investigated for a simple multipath signal environment in which acoustic energy travels along straight line trajectories but with allowance for different effective wave speeds. At sufficient ranges it can be shown that the processor is equivalent to a linear sum of focused beamformers, each steered to a particular multipath. Error in the waveguide steering vectors has the effect of differentially rotating theoretical beams relative to true field beams, resulting in localization bias and peak height reduction. Ambiguous peaks are caused by rotation of a theoretical beam for one multipath onto the true field beam of another multipath. These effects will be discussed with numerical examples.

Effective wave speeds in an SiC-particle-reinforced Al composite

Plane-wave propagation in an SiC-particle-reinforced aluminum-alloy composite was studied. Considering the composite to possess orthotropic symmetry (nine independent elastic constants), by a pulse-echo method, nine independent ultrasonic velocities were measured. Measured elastic stiffnesses departed negatively up to 40% from a rule-of-mixture model. Using ensemble-average, scattered-plane-wave methods, the composite was modeled as SiC particles represented as prolate spheroids distributed randomly, both in position and in orientation. Wave speeds of plane waves, both longitudinal and shear, were calculated in the long-wavelength limit. These wave speeds lead to equations for the effective static bulk and shear moduli of the composite. Further, a nonhomogeneous particle distribution was considered. Wave-speed equations were derived for the case where the composite contains particle-free aluminum-alloy regions that were represented by oblate spheroids. The resulting anisotropic composite behaves transversely isotropicly. For all elastic constants, the model agrees with measurements, within a few percent.

Scattering of elastic waves by spherical inclusions with applications to low frequency wave propagation in composites

Scattering of plane elastic waves by a spherical inclusion is considered. A unified method of solution is presented which treats compressional and shear incidence on a similar basis. Explicit results are given for Rayleigh scattering. We apply the results of the single scattering problem to the propagation of low frequency waves in a composite containing a dilute concentration of spherical inclusions. Explicit formulae are given for the effective wave speeds and attenuations when the inclusions are voids. Both the compressional and shear wave speeds decrease initially as a function of frequency.

Resonance theory of the effective properties of perforated solids

We develop an analytic, unified, and systematic methodology based on scattering theory, for determining the dynamic (frequency-dependent) effective properties of micro-inhomogeneous (visco) elastic media, such as effective elastic moduli, wave speeds, and wave attenuations. The small inhomogeneities are modeled deterministically as fluid-filled spherical cavities, and the Ament–Toksöz method for obtaining static effective properties is extended to the dynamic (resonant) case by retaining those higher-order terms in a long wavelength expansion which give rise to the ultrasonic resonances of the microinhomogeneities. Rescattering from the inhomogeneities has been neglected, thus restricting the applicability of our results to the case of moderate inhomogeneity concentrations. The results of the present analysis are affected by the presence of shear waves in the solid matrix, and they contain all the findings of our previous dynamic resonance theory of gas bubbles in liquids, as well as nearly all the static results available in the earlier literature which do not take rescattering effects into account, as particular cases. We graphically display many numerical results for fluid-filled cavities in rubber matrices.

A Self-Consistent Approach to Multiple Scattering by Elastic Ellipsoidal Inclusions

This paper deals with the scattering of plane longitudinal and shear waves by a distribution of elastic ellipsoidal inclusions. The scattered field is determined correct to O(ε3) where ε is a nondimensional wave number, assumed small. Assuming then that the distribution of scatterer centers is random homogeneous function of position and using a self-consistent (“quasi-crystalline”) approximation effective wave speeds are determined for the case of preferred orientation. Various limiting cases, viz., spherical inclusions and voids, elliptic and penny-shaped cracks, and fluid-filled cavities, are derived.

Elastic wave propagation in a porous laminated composite

An analysis is presented for linear elastic wave propagation normal to the laminations of a periodically laminated porous composite. The porosity is randomly distributed throughout one constituent and is composed of small spherical voids. This type of porosity is called randomly periodic and produces Rayleigh scattering where the wavelength of the incident wave is much larger than the void diameter. Porosity also reduces the wave speed in a constituent and thereby affects geometric dispersion. A dissipative equation of motion is developed for porous material that includes a porosity-dependent wave speed and a scattering term that provides spatial attenuation. This equation is then used for a constituent of a composite, and a dispersion relation and pulse solution are obtained to determine the significance of porosity in a laminated composite. It is concluded that Rayleigh scattering produces a small damping effect in far-field pulse shapes and small-void porosity can be adequately simulated with an effective wave speed.