Journal of Computational Acoustics

The Huygens problem of self-regeneration of the acoustic wave crossing a liquid medium is discussed in the paper. Equal weight of both elastic and kinetic aspects of mechanical waves in fluids is stressed. Two types of virtual surface sources are defined, reflecting local action of the pressure and particle velocity, respectively. They are applied by the author in calculations of secondary radiation from the wave front of the plane wave. The Dirac delta impulse has been used as a waveform, the wave thus being reduced to its own front. The results have been obtained analytically, thanks to some particularly "friendly" features of the operation of convolution with the delta distribution. The paper gives formal proof to Fresnel's intuitive explanation of the mechanism of the forward-only propagation of the wave with no backward effects.

Medical ultrasound imaging can be simulated realistically using linear acoustics. One of the most powerful approaches is to employ spatial impulse responses. Hereby both emitted fields and pulse-echo responses from point scatterers can be determined. Also any kind of dynamic focusing and apodization can be incorporated, as has been done in the Field II simulation program. Here the transducer is modeled through a set of either rectangles, triangles, or bounding lines, so that any geometry can be simulated. The response from the transducer is found by summing the spatial impulse responses from the individual elements. One of the problems in using spatial impulse responses is the abrupt changes in the responses due to the sharp transducer boundaries. Sampling the responses directly therefore have to be done at very high sampling frequencies to keep the shape and energy of the response. The high sampling frequency is unnecessary in the final signals, since the transducers used in medical ultrasound are band limited. Approaches to reduce the sampling frequency are, thus, needed to make efficient simulation programs. Field II uses time integration of the spatial impulse responses using a continuous rather than discrete time-axis. This preserves the energy in the responses and makes it possible to make sub-sample interval delays for focusing. The paper discusses the consequence of the integration for the rectangular elements that uses an approximative calculation of the spatial impulse responses. Data for the accuracy as a function of sampling frequency is given, and it is shown how a sampling frequency of 100 MHz gives similar results to using 2 GHz sampling of the analytic solution for rectangular elements. The spatial impulse responses for the triangular and bounding line elements are found analytically, and an iterative integration routine has to be used. The Romberg integration routine is used, and the accuracy versus sampling frequency for bounding line is shown. An increased accuracy is attained for the lines compared to the rectangles, but the simulation times are significantly higher. Line elements should therefore, in this implementation, only be used very close to the transducer, and if a very high precision is needed in the calculation.

We address the problem of improving the lateral resolution of ultrasonic images by a regularization technique in the wavelet domain. With a very low additional computational cost, the proposed approach increases the efficiency of the standard regularization technique because it efficiently remove the additive noise. Under the assumption that the point spread function is known, we applied our restoration technique to both synthetic and real ultrasonic imaging data. Moreover, experimental results show that the proposed method reduces also speckle artifacts, which generally are enhanced by the deconvolution.

A method for computing the ultrasonic field radiated from segmented annular arrays is presented. Using the well known convolution-impulse response approach, an analytical expression for the spatial impulse response velocity potential from a segmented annular transducer valid for all spatial points is obtained. The array field is then calculated by superposition. The proposed solution allows us to predict transient and continuous wave pressure fields from segmented annular arrays without requiring any far-field or paraxial approximation. In order to evaluate the proposed method, an analysis of efficiency versus accuracy of the computational results is also incorporated.

It is well known that the performances of the acoustic imaging arrays are degraded by the inter-element coupling sustained via the backing, the matching layers, and the kerf filler. The filling material inserted between the elements gives mechanical robustness to the array, but acts as a path of interaction, transmitting, between the elements, shear and lateral compressional forces. In this work, the effect on the array radiation pattern of the cross-coupling due to the filling material is investigated. A hybrid experimental–numerical technique is used. Two groups of five elements of a commercial array transducer were isolated and in one group the kerf filling material was removed. The cross-coupling waveforms, captured with a mechanical probe of small dimensions in contact with the emitting surfaces of the elements, were recorded for filled and unfilled groups of elements when: only the central element of the group was driven; all the elements were driven with the same pulse applied at the same time; all the elements were driven with the same pulse, but inverting the polarity alternately. This latter case refers to the worst coupling situation caused by the shear forces exerted between the elements. Fourier transforming the temporal signals, the cross-coupling transfer function of each element was computed and the radiation pattern was simulated by a numerical model based on the Rayleigh–Sommerfeld integral. Comparing the radiation patterns for filled and unfilled groups of elements, for the three cases mentioned above, a good estimation of the influence of the filling material is obtained.

In this paper, a novel method that is able to synthesize a desired asymmetric beam pattern by a fully optimized linear array to be used together with a narrow-band beamforming system is proposed. Generally, the methods presented in the literature optimize only the weight coefficients and do not consider the element positions. Therefore, when half-wavelength-spaced arrays are concerned, the optimized weight coefficients are valid only for a fixed steering angle. If the steering angle changes, it is necessary to recompute a new set of weight coefficients to keep the desired asymmetric beam pattern profile. The main feature of the method proposed here is that the synthesized weight coefficients keep their validity in any steering direction. This is attained by acting on both the element positions (in order to generate an aperiodic array) and the weight coefficients. To this end, a well-suited energy function has been defined and several conceptual mechanisms have been devised in order to minimize the energy function by the simulated annealing algorithm. As a result, a very flexible method is achieved to synthesize array configurations that yield asymmetric beam patterns that keep their validity in any steering direction without need for any update.

Evaluation of blood-flow Doppler ultrasound spectral content is currently performed on clinical diagnosis. Since mean frequency and bandwidth spectral parameters are determinants on the quantification of stenotic degree, more precise estimators than the conventional Fourier transform should be seek. This paper summarizes studies led by the author in this field, as well as the strategies used to implement the methods in real-time. Regarding stationary and nonstationary characteristics of the blood-flow signal, different models were assessed. When autoregressive and autoregressive moving average models were compared with the traditional Fourier based methods in terms of their statistical performance while estimating both spectral parameters, the Modified Covariance model was identified by the cost/benefit criterion as the estimator presenting better performance. The performance of three time-frequency distributions and the Short Time Fourier Transform was also compared. The Choi–Williams distribution proved to be more accurate than the other methods. The identified spectral estimators were developed and optimized using high performance techniques. Homogeneous and heterogeneous architectures supporting multiple instruction multiple data parallel processing were essayed. Results obtained proved that real-time implementation of the blood-flow estimators is feasible, enhancing the usage of more complex spectral models on other ultrasonic systems.

We discuss a finite-difference modeling technique for scalar, two-dimensional wave propagation in a medium containing a large number of small-scale cracks. The embedding medium can be heterogeneous. The boundaries of the cracks are not represented in the finite-difference mesh but the cracks are incorporated as distributed point sources. This enables the use of grid cells that are considerably larger than the crack sizes. We compare our method to an accurate integral-equation solution for the case of a homogeneous embedding and conclude that the finite-difference technique is accurate and computationally fast.

High-speed trains can generate vibrations that propagate away from the track. We present an accurate and efficient method for computing the vibrations in an elastic half-space at both small and large distances from the track. The method takes possible oscillatory behavior of the train into account. We conclude that vibrations, generated by oscillating trains, can be observed at large distances from the track, even if the train speed is lower than the Rayleigh wave speed.

Sound propagation through a vortex is studied numerically using two different techniques: ray-tracing and finite-differences. Geometrical acoustics and ray-tracing are shown to yield a good picture of the interaction between a sound wave and a vortex when the ratio of the vortex radius to the acoustic wavelength is larger than one. In particular, this technique allows to take into account finite-size effects such as edge waves and the results are compared to experimental data. The interest of the finite-difference approach is demonstrated for cases where sound scattering occurs. We show the ability of such a simulation to account for both sound scattering and finite-size effects. Those two numerical techniques are compared and their validity is investigated.

Low frequency noise (2–200 Hz) reduction in the passenger compartment has emerged in the past few years as a crucial subject of research in the car industry. This kind of noise is mainly due to the panels' vibrations, therefore our aim is to decrease the part of structural energy that reach them, i.e., we want to increase the part of energy that dissipates while propagating in the car body frame. This approach requires the understanding of structural wave propagation through the beam like structure (pillar, cross members…) as well as reflection and transmission at the structural joints. This is the physical problem that we want to address in this paper.

Since car body frames are much too complex for physical understanding, we focused on simpler representative academic structures. We developed a numerical tool for the prediction and visualization of wave propagation, based on finite element models (FEM). Our FEM are first validated by comparison with experimental modal analysis, and then used for transient analysis. In both cases, the good agreement between calculations and experiments shows the reliability of our model and allows us to use it for wave propagation visualization. We illustrate our results by making a movie that helps to understand how waves propagate through a two hollow bodies junction.

A new nonasymptotic method is presented that reveals an unexpected richness in the spectrum of acoustic fluctuations in a shear flow with nontrivial (kinematically complex) mean kinematics. The usefulness of the method is illustrated by analysing three different specific cases of compressible hydrodynamic shear flows. The temporal evolution of perturbations spans a wide range of nonexponential behavior from shear-modified oscillations, transitions between oscillatory and nonperiodic (vortical) modes of motion to monotonic growth. The principal characteristic of the revealed acoustic phenomena is their asymptotic persistence. Exotic regimes like "Echoing" as well as unstable (including parametrically driven) solutions are identified. Further areas of application, for both the method and the new physics, are outlined.

The authors deal with low-cost computer implementation of the integral equation method in application to wave excitation, propagation and diffraction phenomena in solids. Typically these are frequency domain problems of seismic and physical acoustics and of nondestructive testing. Fast codes are obtained basing on analytical extracting and taking account of singular and slowly converging components of the solution beforehand. The present paper gives ideas of several such approaches, which have already proved their efficiency in practical applications. More details including numerical examples and discussion of similar methods known from the literature can be found in the surveyed papers.^1–12

We describe a nonlinear least squares inversion algorithm for obtaining elastic and electromagnetic properties for piezoelectric materials from measured impedances. Richard Brent's PRAXIS, a general unconstrained minimization code is used for the nonlinear least squares fit. No explicit derivatives of the goal functional are required by this code. Bound constraints are imposed in order to limit the variability of the parameters to physically meaningful values. Since PRAXIS is an unconstrained optimization code, these constraints are introduced via a novel change of independent variables. The forward modeling is achieved by using a coupled finite element time domain code for the elastic and electro-magnetic parts of the problem. We also describe how a linearized sensitivity analysis can be used to suggest a priori which parameters can be calculated from impedances measured on a given sample. Numerical results are included.

Several finite element methods for the numerical computation of elastoacoustic vibrations are compared. They are applied to two formulations based on different variables to describe the fluid: presssure and displacement potential in one case, and displacements in the other. While the first one is discretized by standard Lagrangean finite elements for both variables, the second one is solved by "face" Raviart-Thomas elements. In each case we consider both tetrahedral and hexahedral meshes. Elastoacoustic eigenmodes have been computed for a test example by means of MATLAB implementations of all these methods. The numerical results allow us to compare all of them in terms of error versus number of degrees of freedom and computing time.

Ground elastic waves, or ground-borne vibrations, represent one of the major adverse environmental impacts of road vehicles, especially of heavy lorries. In the present paper, ground elastic waves generated by road vehicles are investigated theoretically. Two main generation mechanisms are considered. The first one is associated with vehicles traveling on rough or bumpy road surfaces, in particular over road humps and speed cushions installed by local authorities at some sensitive road locations as a simple method of traffic calming. The second mechanism of generation is associated with acceleration and braking of road vehicles. General analytical results are illustrated by numerical examples and are compared with the existing experiments.

We reconstruct an object which is described by a complex valued function from the scattered waves generated by irradiating plane waves at fixed frequency. The scattering process is modeled by the Helmholtz equation and includes multiple scattering. We present numerical results from computer generated data.

To evaluate the wavefield for realistic 2-D and 3-D models we used a parallel computer, employing algorithms designed specifically to profit from the parallel architecture. The numerical procedures are iterative domain decomposition algorithms employing a nonconforming finite element, which are used to discretize the viscoacoustic and viscoelastic wave equations describing wave propagation in a porous medium saturated by either a single-phase or a two-phase compressible inviscid fluid and subject to absorbing boundary conditions at the artificial boundaries. Our purpose is to establish the effect of gas, brine or oil and gas-brine or gas-oil pore fluids on seismic velocities. Numerical examples showing the implementation of the algorithm to compute crosshole seismic response of simple 2-D and 3-D hydrocarbon reservoirs are presented.

Typical applications of matched-field localization use low frequency signals received on large aperture vertical-arrays. However, small aperture arrays are much more practical for real-world systems and must be considered. Additionally, any practical localization algorithm must also be robust to environmental mismatch. In this paper, we present the broadband L_∞-norm estimator for robust matched-field localization of mid-frequency signals (e.g., 800–4000 Hz) received on very short aperture vertical-arrays. Realistic simulation results are presented using broadband signals in the band of 1000–3000 Hz received on a 3 m vertical array which demonstrate the substantial performance gains in using the L_∞-norm estimator over the asymptotically-optimal maximum a posteriori estimator and the conventional Bartlett processor. Experimental data results in an uncertain shallow-water environment using a 2.13 m vertical array in the band of 3000–4000 Hz are also presented.

Exact solutions for elastic compressional and shear waves scattered from a homogeneous sphere are used to obtain formulas for fluctuations of velocity and attenuation of plane waves propagating through a layer of randomly distributed inclusions over a broad range of frequencies. The size and contrast of the inclusions are arbitrary, but interactions between scatterers are not considered and the concentration of scatterers is assumed to be small. The analytical solutions are also compared with numerical simulations and it is demonstrated that they satisfactorily explain the effects of scattering on both the mean and variance of the phase and the mean and variance of the attenuation. The need for spatial averaging of observational data and methods of interpreting such averaged data in terms of the material properties of the scattering medium are discussed.

In classical mechanics, a time-reversal experiment with a large number of particles is impossible. Because of the high sensitivity to initial conditions, one would need to resolve the positions and velocities of each particle with infinite accuracy. Thus, it would require an infinite amount of information, which is of course out of reach. In wave physics however, the amount of information required to describe a wave field is limited and depends on the shortest wavelength of the field. Thus we can propose an acoustic equivalent of the experiment we mentioned above. We start with a coherent transient pulse, let it propagate through a disordered highly scattering medium, then record the scattered field and time-reverse it: surprisingly, it travels back to its initial source, which is not predictable by usual theories for random media. Indeed, to study waves propagation in disordered media theoreticians, who find it difficult to deal with one realization of disorder, use concepts defined as an average over the realizations, which naturally leads to the diffusion approximation. But the corresponding equation is not time-reversal invariant and thus fails in describing our experiment. Then, to understand our experimental results and try to predict new ones, we have developed a finite elements simulation based on the real microscopic time-invariant equation of propagation. The experimental and numerical results are found to be in very good agreement.

Second-order absorbing boundary conditions for numerical modeling of elastic-wave propagation are studied. The corresponding reflection coefficients are derived, from which a necessary and sufficient condition for complete absorption at normal incidence is deduced. We define a family of absorbing boundary conditions from symmetrically specified zero reflection incidences. Conditions to avoid singular reflection coefficients are given for this case, these ensure that the solutions of the elastic wave equation also satisfy the boundary conditions. These are then optimised over a wide range of materials, and absorbing boundary conditions that give an efficient absorption for the whole range are obtained. We also compare the results with absorbing boundary conditions developed from the least-squares solution of the system requiring complete absorption at all incidences. The best set of conditions are presented and compared with Clayton and Engquist⁶ (A2) condition.

We propose an alternative parameterization of seismic reflectors in the subsurface, in terms of self-similar singularities, which are generalizations of stepfunctions. This parameterization captures the multi-scale behavior of real sonic P-wave velocity logs, as can be derived by performing modulus maxima analysis on wavelet-transformed well-logs. Results on synthetic seismic reflection data, modeled in real well-logs, show that a singularity parameter can be retrieved, that is consistent with the parameter derived directly from the well-log.

In this paper, a method for solving the unknown object problem is presented for a two-dimensional object buried in a fluid seabed lying over a rigid basement. The algorithm makes use of a regularized Born approximation. Several examples are provided which demonstrate the utility of the algorithm.

In this paper, we use a 2-D elastodynamic boundary integral equation or boundary element method (BEM) to solve multiple scattering problems due to existence of cracks or inclusions. The method is based on the integral representation of a scattered wavefield by assuming a fictitious source distribution on the scattering objects or inclusions (i.e. mathematical description of Huygens' principle), and the fictitious source distribution can be found by matching appropriate boundary conditions at the boundary of the inclusions. The method is called indirect boundary element method. Three numerical examples are presented to demonstrate the versatility of the BEM method. The first example shows that different spatial arrangements of the same scatters lead to profound differences in scattering characteristics, in particular the frequency contents of the transmitted wavefields using the method of time-frequency analysis. The second example shows the effects of power-law or fractal distribution of scalelengths on transmitted wavefields, and we conclude that frequency characteristics, such as the frequency of the peak attenuation, can be related to spatial size parameters of the model. In the third example, we show that orientated inclusions with aspect ratio less than unity have strong effects on the amplitudes of transmitted waves, and this has an important implication in characterizing inclusions and fractures using azimuthal variation in amplitudes (or attenuation anisotropy).

We will numerically investigate the wave instability problem with the effect of the transverse velocity component. An accurate and easy to program finite difference scheme will be developed for this purpose. The eigenfunctions will be normalized and computed simultaneously with the eigenvectors. Numerical results will also be presented.

A three-dimensional, two-way parabolic equation model developed for solving three-dimensional backscattering problems in previous work is applied incorporating a focus-marching procedure to study the image reconstructions for underwater targets. Intensive numerical simulations are performed in this paper. Targets in the water are formed by objects with simple shapes. The "experimental data" of the return signals from targets are simulated by the backscattered sound calculated at a short distance from the source. These simulated return signals are processed by reversing their phases, and numerically propagated back to the place of the targets. It is due to the conjugating process that the target image may be reconstructed with less deformation at a distance near the target surface. This realistic study demonstrates an encouraging technique that may be utilized for studying underwater target imaging in many sonar applications.

An exact expression for the reflection coefficient is obtained with the Thomson–Haskell technique for the geoacoustical model of an ocean bottom consisting of an elastic homogeneous sediment layer overlying an elastic half-space. Characteristic equations for explicit determination of the position of each individual resonance contribution to the reflection coefficient are derived. Analytical expressions for the angular and frequency resonance positions are found. The resonance expression for the reflection coefficient is written in the form of a sum of resonance terms. Comparison between resonance theory and exact calculations for the elastic layer covering the elastic half-space is presented. The results of resonance formalism show excellent agreement with exact theory in all the cases.

We present a high-order upwind finite-difference scheme for solving a useful family of first-order partial differential equations, of which the linearized eikonal equation is a member. Fast solutions of the linearized eikonal equation have applications in traveltime tomography and residual migration algorithms. The technique, besides being both accurate and stable, escapes aperture limitations inherent in static marching schemes. We use a time-sequential evaluation method similar to Sethian's Fast Marching strategy to insure causal operator evaluation. We apply our technique to several complex slowness distributions, including the Marmousi model. We also use an adaptation of our technique to compute Cartesian-to-Ray coordinate transforms for the same slowness models.

We compare explicit differential operators for unstructured grids and their accuracy with the aim of solving time-dependent partial differential equations in geophysical applications. As many problems suggest the use of staggered grids we investigate different schemes for the calculation of space derivatives on two separate grids. The differential operators are explicit and local in the sense that they use only information of the function in their nearest neighborhood, so that no matrix inversion is necessary. This makes this approach well-suited for parallelization. Differential weights are obtained either with the finite-volume method or using natural neighbor coordinates. Unstructured grids have advantages concerning the simulation of complex geometries and boundaries. Our results show that while in general triangular (hexagonal) grids perform worse than standard finite-difference approaches, the effects of grid irregularities on the accuracy of the space derivatives are comparably small for realistic grids. This suggests that such a finite-difference-like approach to unstructured grids may be an alternative to other irregular grid methods such as the finite-element technique.

The Elastodynamic Finite Integration Technique (EFIT), originally developed by Fellinger et al.,^1–3 represents a stable and efficient numerical code to model elastic wave propagation in linearly-elastic isotropic and anisotropic, homogeneous and heterogeneous as well as dissipative and nondissipative media. In previous works, the FIT discretization of the basic equations of linear elasticity, Hooke's law and Cauchy's equation of motion, was exclusively carried out in Cartesian coordinates. For problems in cylindrical geometries it is more suitable to use cylindrical coordinates. By that, axisymmetric problems can be treated in a two-dimensional staggered grid in the r,z-plane. The paper presents an EFIT version for axisymmetric problems in cylindrical coordinates called Cylindrical EFIT (CEFIT). After demonstrating the accuracy of the numerical code by a comparison between simulation results and analytical solutions, different examples of application are given. These examples include modeling of sound fields of ultrasonic transducers, thermoelastic laser sources, geophysical borehole probes, impact-echo measurements in layered media, and load simulations of the European Spallation Source (ESS) mercury target.

A general analysis on the subject of acoustic axes for bulk acoustic waves in trigonal crystals is presented. It is shown that the effect of piezoelectricity increases the maximum allowable number of acoustic axes in the plane of elastic symmetry from three to five for trigonal crystals of class 3m, and from 3 to 7 for trigonal crystals of class 32. The theory of acoustic axes of general orientation developed by Khatkevich (1962) is revised. A new, simpler and more general derivation of the conditions for the occurrence of acoustic axes of general orientation in crystals is presented. It is found that the previous analytical results and conclusions stating the absence of such axes in trigonal crystals are incorrect. New correct equations for acoustic axes of general orientation in trigonal nonpiezoelectric crystals are derived and confirmed by independent numerical calculations. Acoustic axes of general orientation oblique to the elastic symmetry planes are found to exist in berlinite (class 32) and in nonpiezoelectric lithium niobate (class 3m), however piezoelectricity eliminates these axes in the latter crystal. Some illustrative examples of the relationship between leaky surface acoustic wave branches and acoustic axes in trigonal crystals are given.

Wave propagation of acoustic waves in porous media is considered. The medium is assumed to have a rigid frame, so that the propagation takes place in the air which fills the material. The Euler equation and the constitutive relation are generalized to take into account the dispersive nature of these media. We show that the connection between the fractional calculus and the behavior of materials with memory allows to work out time domain wave equations, the coefficients of which are no longer frequency dependent.

These equations are suited for direct and inverse scattering problems, and lead to the complete determination of the porous medium parameters.

We design a new and efficient numerical method for the modelization of elastic wave propagation in domains with complex topographies. The main characteristic is the use of the fictitious domain method for taking into account the boundary condition on the topography: the elastodynamic problem is extended in a domain with simple geometry, which permits us to use a regular mesh. The free boundary condition is enforced introducing a Lagrange multiplier, defined on the boundary and discretized with a nonuniform boundary mesh. This leads us to consider the first-order velocity-stress formulation of the equations and particular mixed finite elements. These elements have three main nonstandard properties: they take into account the symmetry of the stress tensor, they are compatible with mass lumping techniques and lead to explicit time discretization schemes, and they can be coupled with the Perfectly Matched Layer technique for the modeling of unbounded domains. Our method permits us to model wave propagation in complex media such as anisotropic, heterogeneous media with complex topographies, as it will be illustrated by several numerical experiments.

I have simulated 0–0.5 Hz viscoelastic ground motion in Los Angeles from M 7.5 earthquakes on the San Andreas fault using a fourth-order staggered-grid finite-difference method. Two scenarios are considered: (a) a southeast propagating and (b) a northwest propagating rupture along a 170-km long stretch of the fault near Los Angeles in a 3D velocity model. The scenarios use variable slip and rise time distributions inferred from the kinematic inversion results for the 1992 M 7.3 Landers, California, earthquake. The spatially variable static slip distribution used in this study, unlike that modeled in a recent study,¹ is in agreement with constraints provided by rupture dynamics. I find peak ground velocities for (a) and (b) of 49 cm/s and 67 cm/s, respectively, near the fault. The near-fault peak motions for scenario (a) are smaller compared to previous estimates from 3D modeling for both rough and smooth faults.^1,2 The lower near-fault peak motions are in closer agreements with constraints from precarious rocks located near the fault. Peak velocities in Los Angeles are about 30% larger for (b) 45 cm/s compared to those for (a) 35 cm/s.

A recently developed theoretical model of the airborne acoustic field from an explosive volcanic eruption of the Strombolian type is described in this article. The magma column is assumed to be a circular cylinder, which is open to the atmosphere at the top, and which opens into a large magma chamber below. The magma itself is treated as a fluid, and the surrounding bedrock is taken to be rigid. An explosive source near the base of the magma column excites the natural resonances of the conduit. These resonances result in displacement of the magma surface, which acts as a piston radiating sound into the atmosphere. The source is modeled in much the same way as an underwater explosion from a high-explosive chemical such as TNT, although in the case of the volcano the detonation mechanism is the ex-solution of magmatic gases under extremely high hydrostatic pressure. The new theory shows compelling agreement with airborne acoustic signatures that were recorded in July 1994 at a distance of 150 m from the western vent of Stromboli volcano, Italy. The theoretical and observed power spectra both display the following features: (1) four energetic peaks below 20 Hz, identified as the first four longitudinal resonances of the magma column; (2) a broad minimum around 30 Hz, interpreted as a source-depth effect, occurring because the source lay close to nulls in the fifth and sixth longitudinal resonances and thus failed to excite these modes; and (3) radial resonance peaks between 35 and 65 Hz. On the basis of the theory, an inversion of the acoustic data from Stromboli yields estimates of the depth (≈100 m) and radius (≈16 m) of the magma column as well as the depth (≈83 m), spectral shape and peak shock wave pressure (≈1 GPa) of the explosive source. Most of the parameters estimated from the acoustic inversion compare favorably with the known geometry and source characteristics of Stromboli.

Linear theory is developed of flexural wave motions of an elastic beam of circular cross-section traveling along its axial direction at a constant speed in an air-filled, circular tube placed coaxially. The beam is constrained to deflect in a plane and is subjected to a restoring force in proportion to the magnitude of deflection. Both beam and tube are assumed to be long enough for end effects to be ignored. Taking account of aerodynamic loading on the lateral surface of the beam, wave propagation is examined under a long-wave approximation that a characteristic wavelength of flexural waves is much longer than the tube's radius. Using the lossless, acoustic wave equation for the velocity potential of the air, and the classical, flexural wave equation for the deflection of the beam, an asymptotic-expansion method is applied based on the long-wave approximation. A linear wave equation for the deflection is derived and its dispersion relation is examined by comparing with the exact one derived previously without any long-wave approximation. Higher-order effects of the axial curvature of the beam and the compressibility of air on the aerodynamic loading are also discussed.