The algorithm Swamp has been benchmarked and documented and has been effectively used in a variety of studies [see http://www.ucs.louisiana.edu/∼nxs7560]. Here the method is augmented in two ways. First, a more general layering scheme is implemented which allows for a broad angular spectrum required to replicate near field effects. Second, by making use of pseudo unitary operators the method is extended to a two-way coupled mode algorithm. The method is tested and compared with available methods in current use. Further, it is possible to make use of bandedness and this improves the time of calculation significantly and allows the resulting code to be used for very high frequencies. Benchmark calculations are presented for several examples.

The T-matrix method initiated by Waterman was extended in a clever paper by Peterson and the Varadans in 1980 to the complicated problem of elongated elastic shells. There have been some questions on whether the method converges for thin shells due to the impression that one must be able to inscribe a spherical surface in the annular region of the shell. The method may be derived by making use of several of the constraining equations that arise naturally and we show that the expression derived in 1980 is generally correct at least for objects with mirror symmetry. We present some details of the theoretical development with some calculations.

For a known source and suitable environmental information one may determine the presence (or absence) of a target ensonified by a guided wave initiated by the source by using vertical arrays that sandwich a region of interest. We show first that one can represent the ensonified object as a source term in the solution of an inhomogeneous Sturm–Louiville problem where the homogeneous solution corresponds to the solution in the absence of a target. This allows one to choose a bounded region about the object and evaluate the equivalent of Huygens’ integral (over the boundary that includes the vertical arrays), which enables one to trace the signal back to the object. This determines both the range and depth and even the target strength. When this is extended to a pulse signal then the information leads to higher fidelity than the frequency results due to phase averaging of fluctuations. Thus we demonstrate this to be a good strategy for an experimental study that would lead to a robust detection method that may be implemented on a rapid basis.

Initially let a pulse signal progress in time in a wave-guide along one direction ab initio at time t0. If it interacts with a submerged structure, at some time t1 then part of the wave will be refracted and part will be reflected so that at some point along the range between the object and the source there will be interference between the pulse signal and the reflected (backscattered) signal. This, to be sure, is a function of the duration of the pulse as well as other factors. Thus, in that region we must be required to deal with a two-way solution no matter what the topography simply because of the interference between the reflected wave and the time evolving wave. Here, the most general problem is properly formulated and the effect is examined as a function of pulse width, frequency (band) width, gating, and other factors. Results are presented in the form of frames or animations and it is observed that one may gain insight into this complicated process and it is possible to make use of this effect for extracting inverse information. [Work supported by ONR and NRL.]

Water-filled elastic shells present more complicated backscattered echoes when compared to evacuated shells and are less well understood. The objective here is to determine and explain features peculiar to such objects and to use these features to aid in object characterization. Towards that end, an acoustic background suitable for such targets in conjunction with "residual" partial wave analysis obtained from subtracting the background from the elastic response is used. This analysis is aided by the determination and comparison of plate modes and resonance locations of evacuated unloaded, evacuated loaded, and loaded water-filled shells. Analysis of water-filled shells suggests that there exist isolated narrow, uniformly spaced eigenmodes in addition to the elastic modes. The broader elastic modes appear split in comparison to their unloaded counterparts. Otherwise, the elastic modes are not greatly affected by the included eigenmodes. The isolated (included) eigenmodes are extremely well modeled by a water inclusion in an infinite elastic matrix composed of the elastic shell material. The splitting of elastic modes proves to be due to interference of the included eigenmodes and the elastic body resonances. This leads to what appear to be separate branches of dispersion curves related to the elastic modes. Because of the numerous and usually evenly spaced included eigenmodes, dispersion curves are dominated by the abundance of these modes. A clear picture of the physical processes emerges from this analysis that explains all features of this event. Simple sets of rules that lead to tractable calculations are introduced that facilitate analysis of this interesting physical process.

Exact equations exist for the determination of the backscattered signals excited by acoustic signals on submerged elastic shells. In the absence of resonances these signals may be cast in terms of conservation principles of inertial components of the interaction of the signal with the object. This indeed leads to an adequate description of the acoustic background of such targets. What is determined to be critical is the inclusion of an entrained mass related to the displacement of the mass equivalent of the fluid due to the movement of the object. But what happens to such inertial effects in the presence of elastic resonances? We explore this issue and demonstrate that the effective or inertial component ranges from zero to infinity over a small frequency range that characterizes the resonance width. Over this range there is also a phase shift of 1800. We may as well examine the effect of radiation loading in this range. It is also possible to investigate a phase plot of the real and imaginary components of the effective mass factor which leads to closed trajectories for certain classes of resonances and to hyperbolic trajectories for other types. The meaning of these observations is discussed. [Work supported by NRL and ONR.]

A comparative study is presented of the acoustical excitation of circumferential (surface) waves on fluid-immersed cylindrical or spherical metal shells, which may be either evacuated, or filled with the same or a different fluid. The excited surface waves can manifest themselves by the resonances apparent in the sound scattering amplitude, which they cause upon phase matching following repeated circumnavigations of the target object, or by their re-radiation into the external fluid in the manner of head waves. We plot dispersion curves versus frequency of the surface waves, which for evacuated shells have a generally rising character, while the fluid filling adds an additional set of circumferential waves that descend with frequency. The resonances of these latter waves may also be interpreted as being due to phase matching, but they may alternately be interpreted as constituting the eigenfrequencies of the internal fluid contained in an elastic enclosure.

This paper considers acoustic scattering from submersed, elongated elastic objects, i.e., solid spheroids and finite-length cylinders. Surface waves are generated on the objects by incident acoustic waves; they can cause resonances in the objects’ vibrations which become evident in the scattered acoustic amplitudes. As it has been shown previously, the resonance frequencies can be obtained by a phase matching condition for multiply circumnavigating surface waves as they propagate on the object over closed paths. Comparing with calculated resonances in the scattering amplitude, it is found that the phase matching condition furnishes correct values for the calculated resonance frequencies for the case of Rayleigh-type surface waves. For spheroids and cylinders of larger aspect ratio, it is shown that bar-wave induced eigenfrequencies may also be used to approximate the calculated resonance frequencies.

Acoustic signals scatter from an elastic shell and excite elastic shell resonances. For evacuated shells resonances due to proper Lamb waves, A0 and S0 waves and the pseudo-Stoneley resonances, are the only ones allowed. When the shell is fluid filled then the eigenfrequencies of the included fluid may be excited. Further, the presence of the fluid may alter the elastic body resonances. When the impedance of the entrained fluid is not much smaller than the mechanical impedances of the elastic material then the scattered signal is greatly influenced by the entrained fluid, but only at their allowed eigenfrequencies. Since the fluid is entrained this leads to a discrete spectrum and an eigenvalue problem which is quite manageable. In this work we outline a method for determining the eigenfrequencies as well as their nature and use their values to isolate the actual body resonances due to the elastic material. We also illustrate the influence of the presence of the entrained fluid on the elastic resonance frequencies. [Work supported by NRL and ONR.]

Some oceans appear to have layers that are less dense and have slower speeds than overlying layers. Perhaps this is due to a gaseous admixture with some of the sediment. This presents some interesting effects in model calculations which are dependent on frequency, layer thickness, and adjacent layers. The mathematical consequence of this is examined.

If one illuminates the ocean with a time-gated acoustical signal will an inclusion such as an elastic shell insonfied by the pulsed twinkle or, in the jargon of some researchers, will it scintillate? Twinkling from stars owes its origin to a mildly turbulent atmosphere. The ocean is often much more turbulent and we examine the distortions that one gets from scattered pulse-gated signals.

Negative group velocities have been reported in calculations for waves generated on elastic objects. It is usual to interpret a group velocity with the travel time of a wave front or the time that the energy of a wave generated travels along some trajectory. One may look at the group velocity as the rate at which an envelope of a wave train travels. More generally, one may exploit the principle of stationary phase along with the Fourier transform in frequency and derive the concept of group velocity from the first order asymptotic term. This derivation requires that the modulus of the integral is changing slowly relative to the phase term. In addition it is required that the group velocity is not near a stationary point in frequency. In that case first order asymptotics is no longer valid and one must employ higher asymptotics. Here, the standard concept of group velocity is no longer meaningful. Further, the signal becomes amplitude modulated rather than frequency modulated. We examine this for some examples, assuming losslessness. In the event of loss the concept of group velocity has already been reported to be of dubious value.

The onset of the excitation of flexural resonances for fluid loaded evacuated elastic shells produces a striking event. This issue has been interpreted theoretically in 1988 by means of a partial wave decomposition which showed a very narrow-peaked subsonic wave and a broad-peaked weak wave that begins at the speed of sound of the entraining fluid and increases. The narrow peaks are identified with subsonic water-borne waves that resonate in the fluid along the surface of an elastic object, and the broad peaks correspond to the inception of flexural waves. They exist over a small interval in wave number at the point when the flexural wave begins to couple with the fluid. Here we examine the pulse solution. With increasing frequency the partial waves change phase (a 90 degree phase change) and the overlapping flexural waves transition from a partially coherent constructive signal to one that is partially destructive. This leads to an envelope or hump effect, also called mid-frequency enhancement; it is a function of shell thickness as well as material property. We demonstrate how this effect may be employed to identify a submerged elastic shell in either the pulse or frequency solution.

Results derived from exact linear homogeneous elastodynamic theory are used for two-dimensional unloaded plates in order to understand certain features generated by proper symmetric Lamb modes. It is shown that S1 modes for all elastic materials have a phase velocity defined below the usual critical frequency and which initially exhibits anomalous dispersion (has a negative slope with respect to frequency). Over a certain range, it has a phase velocity that is double valued. In addition, there are an infinite number of proper symmetric Lamb modes that have this characteristic for materials with a Poisson ratio equal to 1/3. It also appears that all A3n modes are anomalous when V(L) < or = 2 V(T). The cause and implication of these effects are examined, including an associated negative group velocity over a small frequency zone for these modes. Further, it is noted that all proper symmetric Lamb modes have a plateau region in phase velocity with respect to wave number. It is shown that this always occurs for a phase velocity corresponding to the longitudinal bulk velocity of the elastic material. These issues are examined along with how one may obtain material parameters and possibly plate thickness from their dispersion curves.

Resonances excited on any shell of constant thickness are one large resonance at a frequency inversely proportional to the shell thickness. Hence for thin shells it occurs at very high frequency. This effect occurs at the inception of the S1 resonance, which may be shown to be an amplitude-modulated wave at inception. Its critical frequency may be determined by the condition: ka=3.14 VLa/(2 daVW) or ka=3.14 VTa/(daVW) where VW, VT, VL, a, da, and k are the speed of sound in the ambient fluid, transverse and longitudinal velocities in the elastic material, largest dimension of the object, object thickness relative to a and the wave number in the fluid. If 2VT>VL the first condition defines the S1 critical frequency and the second that of the S2. The converse is true otherwise. The S2 resonance is not striking but may be identified as such. Thus, ratios of the two resonances lead to ratios of the bulk velocities and other considerations can isolate the shell thickness. This offers for any shell of constant thickness a means to determine the presence of certain submerged objects. We discuss the reasons for this and illustrate results in both time and frequency domains.

A large class of resonances is observed from the excitation of elastic shells. A review of the type of resonances is presented. It is then shown how to label the resonances depending on material properties, ambient fluid, shape of object, and evacuation or fluid filling.

Assume one has knowledge of the Green’s function in some ocean waveguide and a measurable signal scattered from a source. This may be treated by solving the Fredholm equation of the first kind (FEFK). The task is to determine any features from the measured field by solving the FEFK. We use synthetic data and produce measurents and determine what information may be recovered by solving this equation.

The second order solution of the wave equation is useful in adopting a method for solving the related inhomogeneous version in which the inhomogeneous term comes from a known target or an unknown object in the waveguide. Formulations are presented for these cases with examples.

Acoustic resonance spectra are calculated for scattering of acoustic signals from elongated objects composed of six elastic materials with two aspect ratios of 3 and 6. The incident field is along the axis of symmetry and broadside. A comparison is made of the resonance locations of the six materials (brass, nickel, aluminum, steel, manganese, and tungsten carbide) representing a broad spectrum of elastic materials. For the principal resonances the ratios are seen to correspond exactly to the Rayleigh phase velocities on an evacuated half-space, or alternatively to the shear bulk velocity and are a function of the Poisson ratios of the material. Further, the resonance widths are related inversely to the density of the material and the shear velocity (the mechanical impedance of the shear wave). Time-domain calculations are also carried out and the resonance widths and travel times may be identified with the material properties of the target. Thus, the material properties of such objects including elongation may be distinguished for submerged objects, and this is a useful tool for inverse issues.