Subsonic Flexural Waves Research Articles

Modulation of the vibroacoustic behavior of a partially immersed cylindrical shell by regular waves

The effect of small regular waves on the vibroacoustic behavior of a partially immersed cylindrical shell under point-force excitation is studied using theoretical and experimental approaches. An approximate theoretical model of the circumferential vibroacoustic behavior of this system is established by assuming that the waves only change the draught of the shell. Vibroacoustic coupling equations are derived, and experimental verification is carried out in a towing tank. Both the theoretical and experimental results show modulation of the vibration and sound radiation of the shell by the water waves. These modulation effects are determined by the wave parameters, namely, (i) the fluctuation period of the resonant peaks in the radial velocity and sound pressure is determined by the wave period; (ii) the bandwidth of the modulation is determined by the wave amplitude. For a partially immersed cylindrical shell subjected to regular waves, the air–water demarcation points on the shell surface are the sources of the acoustic radiation. This radiation is modified by the water waves, and the propagation and radiation paths of the subsonic flexural waves and the resonant frequency of the shell are changed as a result. Based on this mechanism, a simple formula is derived to predict the modulation of the resonant frequency of the radial velocity and the sound pressure using the phase velocities of the subsonic flexural waves on the dry and wet parts of the shell. The predicted results are consistent with the theoretical and experimental results.

Acoustic scattering from an infinitely long cylindrical shell with periodic internal lengthwise ribs.

The acoustic scattering from an infinitely long cylindrical shell with periodic lengthwise ribs is studied. The shell motion is described by the Donnell equations, and the lengthwise rib is modeled as an elastic beam whose motion is decomposed into longitudinal and flexural vibrations. A circumferential mode expansion is used to obtain numerical results for the scattering sound field. The backscattering characteristics in the far-field can be explained by the resonance and interference phenomena. It is shown that subsonic flexural waves can be generated and radiated by the ribs. Due to the periodical distribution of the ribs, there exist multi-order flexural Bloch waves in the circumferential direction. The multi-order flexural Bloch waves can form standing circumferential waves, which lead to a complex acoustic resonance. The attachments of the ribs to the shell can reflect an acoustic wave directly and the reflected wave will interfere with the specular reflection, which is dominant in the frequency-angle spectra with an increasing number of ribs. Furthermore, the flexural wave and flexural Bloch waves can radiate through the attachments and interfere with the specular reflection. However, the interference fringes in the frequency-angle spectra caused by the flexural wave and flexural Bloch waves are coincident at broadside.

Vibroacoustic behavior of a partially immersed cylindrical shell under point-force excitation: Analysis and experiment

The vibroacoustic behavior of a partially immersed cylindrical shell under point–force excitation is studied using theoretical and experimental approaches. An analytical form of the vibroacoustic coupling equation is developed using a discretization approach for the circumferential angle along the shell’s wet surface. The radiated sound and radial velocities are analyzed using frequency–depth spectra, and a series of regular oblique bright lines and weak interference fringes can be observed. For a partially immersed cylindrical shell, the subsonic flexural wave a0 can radiate energy into the fluid from air-fluid demarcation points on the shell surface, and produce a series of resonant bright lines in the pressure spectra when the shell resonates in the circumferential direction. Interaction between the radiated waves, which propagate on parts of shell above and below the free surface (on the “dry part” and “wet part”, respectively), produces interference fringes in the frequency–depth spectra of sound pressure. Furthermore, simple formulas are given to predict the lines and fringes using the phase velocities of a0 wave on the dry and wet parts of the shell. Finally, experimental verification is carried out, and the measured frequency–depth spectra of the surface velocity and the sound pressure are consistent with the theoretical results.

Acoustic reflection and transmission by sub-critical and super-critical elastic partitions with structural discontinuities forced by multiple-angle oblique broadband sound waves

An analysis is presented for acoustic reflection and transmission from an infinite fluid-loaded flexible barrier with spatially periodic discontinuities. The plate, with similar or dissimilar acoustic fluids on both sides, is excited by an oblique wave incident acoustic field. The fully-coupled structural/acoustic problem is treated by Analytical-Numerical Matching (ANM) which improves the numerical accuracy and convergence rate. Periodic spatial discontinuities, modelled by various boundary conditions, create deviations from specular directivity. For super-critical structural wave speeds, the scattering is characterized by a distribution of reflection and transmission angles centered around the Mach Angles. For sub-critical structural waves, energy redistribution is most pronounced at the structural resonances. These results are compared to a baffled finite barrier analyzed by approximate means. For the subcritical finite barrier, a fluid loading correction is introduced to the structural wavenumber to approximate the effects of the fluid coupling and radiation damping. For subsonic flexural waves, the radiation is analyzed in terms of the wavenumber transform and interpreted as a series of edge singularities dependent on the boundary conditions. For the super-critical configuration, the effects of the fluid loading are introduced by approximating the structural wavenumber from the dispersion relation of the free response of the unbounded structure.An analysis is presented for acoustic reflection and transmission from an infinite fluid-loaded flexible barrier with spatially periodic discontinuities. The plate, with similar or dissimilar acoustic fluids on both sides, is excited by an oblique wave incident acoustic field. The fully-coupled structural/acoustic problem is treated by Analytical-Numerical Matching (ANM) which improves the numerical accuracy and convergence rate. Periodic spatial discontinuities, modelled by various boundary conditions, create deviations from specular directivity. For super-critical structural wave speeds, the scattering is characterized by a distribution of reflection and transmission angles centered around the Mach Angles. For sub-critical structural waves, energy redistribution is most pronounced at the structural resonances. These results are compared to a baffled finite barrier analyzed by approximate means. For the subcritical finite barrier, a fluid loading correction is introduced to the structural wavenumber to a...

Experimental analysis of backscattering by cylindrical shell with internal plate at oblique incidence

Through an experimental approach, this paper studies the flexural wave coupling on a cylindrical shell with internal structural. Impulse response backscattering measurements are presented and interpreted for the scattering of obliquely incident plane waves by a fluid-loaded stiffened cylindrical shell and a corresponding empty cylindrical shell respectively. The stiffened cylindrical shell is reinforced by a thin internal plate which is diametrically attached to the shell along its axial direction. The time series data are fast Fourier transformed and the modulus normalized according to the direct wave spectrum. Result are plotted as frequency-angle spectra. Compared with that from the corresponding empty cylindrical shell, the subsonic flexural waves on the cylindrical shell will interact between the attachments and some of their energy can be converted into radiating waves.

Tunable cylindrical shell as an element in acoustic metamaterial.

Elastic cylindrical shells are fitted with an internal mechanism which is optimized so that, in the quasi-static regime, the combined system exhibits prescribed effective acoustic properties. The mechanism consists of a central mass supported by an axisymmetric distribution of elastic stiffeners. By appropriate selection of the mass and stiffness of the internal mechanism, the shell's effective acoustic properties (bulk modulus and density) can be tuned as desired. Subsonic flexural waves excited in the shell by the attachment of stiffeners are suppressed by including a sufficiently large number of such stiffeners. The effectiveness of the proposed metamaterial is demonstrated by matching the properties of a thin aluminum shell with a polymer insert to those of water. The scattering cross section in water is nearly zero over a broad range of frequencies at the lower end of the spectrum. By arranging the tuned shells in an array the resulting acoustic metamaterial is capable of steering waves. As an example, a cylindrical-to-plane wave lens is designed by varying the bulk modulus in the array according to the conformal mapping of a unit circle to a square.

Scattering of plane evanescent waves by cylindrical shells and wave vector coupling conditions for exciting flexural waves

The coupling of sound to buried targets can be associated with acoustic evanescent waves when the sea bottom is smooth. To understand the excitation of flexural waves on buried shells by acoustic evanescent waves, the partial wave series for the scattering is found for cylindrical shells at normal incidence in an unbounded medium. The formulation uses the simplifications of thin-shell dynamics. In the case of ordinary waves incident on a shell, a ray formulation is available to describe the coupling to subsonic flexural waves [P. L. Marston and N. H. Sun, J. Acoust. Soc. Am. 97, 777–783 (1995)]. When the incident wave is evanescent, the distance between propagating plane wavefronts is smaller than the ordinary acoustical wavelength at the same frequency and the coupling condition for the excitation of flexural waves on shells or plates is modified. Instead of matching the flexural wave number with the propagating part of the acoustic wave number only at the coincidence frequency, a second low-frequency wave number matching condition is found for highly evanescent waves. Numerical evaluation of the modified partial-wave-series appropriate for an evanescent wave is used to investigate the low-frequency coupling of evanescent waves with flexural wave resonances of shells.

Elastic wave power equipartition in a finite-length ringed shell

Compressional and shear waves, trace matched on a submerged cylindrical shell by plane sound incidence near beam aspect, can couple to each other, as well as to subsonic flexural waves, at ring stiffeners and endcaps. This paper addresses whether the power in each elastic wave type (defined as energy per unit shell length multiplied by the wave axial group speed) is partitioned equally with the other two. Shell surface velocities of a ringed shell, calculated for midfrequency range 2<ka<12 via a SARA 2-D program implemented by Liu, are Fourier transformed into the wave-number domain to evaluate each wave type, and then into the time domain to evaluate transient wave power in several time windows. The time windows are based on an integral number of roundtrips of the trace matched wave in the shell. It is found that elastic wave power among the wave types differs by 10 dB in the first window, and therefore is not equipartitioned. However, the wave power difference is within 3 dB in the second window, which supports the conclusion of at least approximate wave power equipartition. [Research supported by ONR.]

Acoustic interaction with wedge-shaped structures

This paper will review recent developments in solving canonical structural acoustics problems involving wedgelike structures composed of plates and membranes. The two cases considered are (i) a membrane held taut over a line constraint, and (ii) two semi-infinite thin plates welded together to form a wedge. The wedge-shaped structures are subject to unilateral fluid loading from an acoustic fluid. The method of Malyuzhinets, which was used in the 1950s for the purpose of dealing with simple impedance-type boundary conditions on the wedge faces, has recently been significantly developed by A.V. Osipov, who has shown how it can be successfully applied to these structural acoustics problems. The central idea of the Malyuzhinets/Osipov method is to express the acoustic pressure using a generalized Sommerfeld integral, similar to a Laplace transform. The main analytical steps in the solution will be discussed, and some numerical results will be presented. The plate structure is particularly interesting because both subsonic flexural and nonleaky supersonic structural waves are excited at the vertex by an incident acoustic wave. Interesting asymptotics emerge when the wedge angle described by the fluid is almost 180o or 360o. [Work supported by ONR.]

Ray acoustics and vibration of immersed thin elastic cylindrical shells excited by an on-surface source

Structural loadings, joints, and truncations of smooth shells act as equivalent forcings on such shells in the overall acoustic response and vibration. Choosing an on-surface phased line source as the acoustic source, this paper treats the ray synthesis of the pressure and displacement fields on a thin elastic cylindrical shell immersed in an acoustic fluid as well as the sound field in the far zone. The subsonic flexural waves and their strong resonances are observed at the shell surface, but not in the far field, thereby generating distinct field structures in these two regions. In the parameter region where the supersonic compressional and shear waves are cutoff, the surface fields are found to be represented completely by the shell guided flexural mode, while the far field is reduced to the modified geometrical acoustics field (essentially comprised of incident and reflected waves). The ray acoustic predictions prove to be accurate in comparison with the normal mode series data for ka ⩾ 2 or so (where a is the mean radius of the shell and k is the acoustic wavenumber of the fluid).

Energy branching of a subsonic flexural wave on a plate at an air–water interface. I. Observation of the wave field near the interface and near the plate

The radiation of subsonic flexural plate waves due to a discontinuity in fluid loading is experimentally investigated. A tone burst of flexural waves propagates down a plate, the lower section of which is submerged in water. Measurements indicate that there occurs a branching of energy as the flexural wave passes through the air–water interface, with little reflected energy. A portion of the transmitted energy continues along the plate as a subsonic flexural wave with an associated acoustic evanescent wave. A second acoustic wave (which is termed transition radiation) originates at or near where the plate crosses the interface, and propagates in water to the far field. In the near field of the interface there exists an interference between the two acoustic waves in water that results in a series of pressure nulls. The pressure nulls are associated with a π phase change in the wave field and are indicators of wavefront dislocations. A numerical computation of the wave field in an unbounded fluid due to a line-moment excitation of a plate has similar features as the null pattern observed but differs in certain details.

Scattering from flexural waves on a ribbed cylindrical shell

Monostatic scattering measurements over a ka range 2–30 made on two cylindrical shell structures, one with regular internal frames and one without, clearly demonstrate that the periodic discontinuities in the framed shell give rise to distinctive features in the scattering patterns at frequencies much lower than those at which Bragg scattering peaks can exist. These features are found to be even more dominant than those associated with the now well-known scattering from supersonic shear and compressional helical shell waves. When the measured scattering levels are displayed as a function of frequency and aspect, these features manifest themselves as curves suggestive of folded-over dispersion curves of flexural waves. It will be shown that these effects are, in fact, due to scattering from Bloch or Floquet wave packets arising from the frame-induced multiple scattering of the subsonic flexural waves. The measured scattering functions are compared with predictions based on phase matching of the acoustic waves to flexural Bloch waves whose dispersion properties are computed using an approximate model. The simple model predictions agree well with the measured scattering data.

Generation, diffraction, and radiation of subsonic flexural waves on membranes and plates: observations of structural and acoustical wave fields

Generation, diffraction, and radiation of subsonic flexural waves on membranes and plates: observations of structural and acoustical wave fields

Reflection of a subsonic flexural wave on a plate at an air–water interface and far-field observations of the transition radiation in water

An investigation of radiation by subsonic flexural plate waves due to a discontinuity in fluid loading is described. A tone burst of flexural waves propagates down a plate, the lower section of which is submerged in water. Measurements indicate that there occurs a branching of energy as the flexural wave passes through the air–water interface, with little reflected energy. A portion of the transmitted energy continues along the plate as a subsonic flexural wave with an associated evanescent wave. A second acoustic wave (which is referred to here as term transition radiation) originates at or near where the plate crosses the interface, and propagates in water to the far field. The near-field interference between the two acoustic waves in water results in a series of pressure nulls [T. J. Matula and P. L. Marston, J. Acoust. Soc. Am. 94, 1877 (A) (1993); T. J. Matula, Ph.D. dissertation (1993)]. Far-field observations of the transition-radiation angular pattern resemble that of a line quadrupole (approximating the farfield radiation due to a line moment on an uniformly loaded thin plate). The agreement between measured and modeled patterns is improved, however, by displacing the quadrupole horizontally behind the surface of the plate. [Work supported by ONR.]

Active Control Of Sound Radiation From A Semi-Infinite Elastic Beam With A Clamped Edge

Active control of sound radiation due to subsonic wave scattering from a clamped edge discontinuity on a thin semi-infinite beam is analytically studied. Active control is achieved by applying either control forces representative of shakers or paired control moments representative of a piezoelectric actuator bonded on the beam. For a single frequency, the flexural response of the beam subject to an incident subsonic flexural input wave and the control inputs is expressed in terms of waves of both propagating and near-field types. The optimal control input complex amplitudes are derived by minimizing the acoustic power radiated from one side of the baffled beam. For single frequencies, large attenuation in radiated acoustic power and pressure is found when one and two control forces or one pair of control moments are applied close to the discontinuity. The far-field radiated pressure directivity, the displacement of the vibrating beam, and the one-dimensional wavenumber spectrum of the beam velocity are extensively studied in order to further understand the control mechanisms. In general, large attenuation in radiated sound are shown to be associated with only very small changes in beam vibrational response near the discontinuity.

Energy branching of a subsonic flexural wave on a plate at an air–water interface: Transition radiation and the acoustic wave field in water

The diffraction of subsonic flexural plate waves due to a discontinuity in fluid-loading is experimentally investigated. A tone burst of flexural waves propagates down a plate, the lower section of which is submerged in water. Observations indicate that there occurs a branching of energy as the flexural wave passes through the air–water interface. A portion of the energy continues along the plate as a subsonic flexural wave with an associated evanescent wave. A second acoustic wave (which is termed transition radiation) originates at or near where the plate crosses the interface, and propagates in water to the far field. In the near field of the interface there exists an interference between the two acoustic waves in water that results in a series of pressure nulls. The pressure nulls are associated with a π phase change in the wave field and are indicators of wave front dislocations [P. L. Marston, ‘‘Geometrical and Catastrophe Optics Methods in Scattering,’’ Physical Acoustics (Academic, New York, 1992), Vol. 21, pp. 1–234]. A computation of the wave field in an unbounded fluid due to a line-moment excitation of a plate is comparable with the null pattern observed but differs in certain details. [Work supported by ONR.]

On the acoustic advantages of a helical‐ring stiffener in a thin cylindrical shell

A conventional circular‐ring stiffener in a thin cylindrical shell may cause undesirable sound radiation due to the scattering of subsonic flexural waves. Vibration in each of a set of disjointed circular rings may stay within the ring, which results in a resonant buildup that acts as a localized excitation of the shell. Furthermore, elastic waves in a cylindrical shell tend to propagate, having a plate‐wave front, along the axis of the cylinder. A circular ring is oriented in the plane normal to the direction of wave propagation and therefore it sees the elastic wave as a normal incident wave. These rings are separated at a certain distance in tandem. An incident wave in the shell may be reflected back and forth between a pair of adjacent rings. Therefore, a circular ring may not only be an efficient scattered, it could also scatter the same wave several times causing a reverberant buildup in the shell between rings, and, consequently, enhance acoustic radiation from the shell. This paper discussed the technical rationale to alleviate these shortcomings by using a continuous helical‐ring stiffener in a thin cylindrical shell.

Scattering of impulse signal by elastic shells with internal structures.

Sound scattering from an elastic shell can be significantly affected by structures loaded inside the shell. A major effect of internal loading has been shown to be the conversion of subsonic flexural waves in the shell into strongly radiating acoustic waves. In the frequency domain, this leads to a deeply scalloped scattering form function with the empty-shell result as its ‘‘mean value.’’ Since flexural waves in the shell are much slower than supersonic compressional leaky waves that dominate the scattering process for an empty shell, the conversion of flexural waves into sound by internal structural loading is well separated in time from other processes such as the initial ‘‘bubble-like’’ reflection and the phase-matched leaky wave re-radiation. Also, because of the dispersive nature of the flexural waves, the internal loading effect in time domain can be clearly identified; they distinguish themselves in the scattered field by a series of wave packets whose amplitudes decay due to both acoustic damping and dispersion. This is in contrast to the other two main components in the scattered time signal, one being the initial reflection that is basically the inverse of the incident signal and the other being the leaky wave contribution consisting of decaying pulses. All these are clearly demonstrated in this paper, which examines the scattering of impulse incidence from elastic shells with internal structures. The response of the loaded shell is also examined, which further reveals the interactions between the acoustic field, the shell, and its internals. [Work supported by ONR.]

Three-dimensional scattering from a rough fluid-elastic interface

Out-of-plane scattering is of interest to the Arctic noise community because of the observation of significant out-of-plane motion in ice excited by explosive sources in the water column. Because low-frequency fluid-elastic scattering apparently couples energy into subsonic flexural waves as well as horizontally polarized shear (SH) waves in the ice, radiating in all directions, some impact on Arctic transmission loss and horizontal coherence properties is anticipated. The self-consistent boundary perturbation formulation for rough surface scattering developed by Kuperman and Schmidt [J. Acoust. Soc. Am. 86, 1511–1522 (1989)] is used to estimate the mean and scattered fields for a two-dimensionally rough fluid-elastic interface excited by an incoming plane wave. The reflection and scattering coefficients determined by the full 3-D formulation are compared to earlier results, where the surface roughness was assumed to be polarized in the plane of the exciting wave only. In addition, the kernel of the wave-number integral for the scattered field is examined and it is found that significant energy is scattered into Scholte waves as well as SH waves at all azimuths relative to the incoming plane wave, consistent with the experimental observations. [Work supported by ONR.]

Low-frequency diffraction from a free surface coupled to a semi-infinite elastic surface as modeled by sea ice properties

This paper discusses the solution of a low-frequency plane wave incident upon a semi-infinite elastic plate, such as an Arctic ice lend or free edge, using the Wiener-Hopf method. By low-frequency it is meant that the elastic properties of the plate are adequately described by the thin plate equation. For example, in a floating ice sheet, this translates into frequency-ice thickness products that are ≲ 150. A key issue here is the fluid loading pertaining to sea ice and low-frequency acoustics, which cannot be characterized by simplifying heavy or light fluid loading limits. An approximation to the exact kernel of the Wiener-Hopf functional equation is used here, which is valid in this midrange fluid loading regime. The farfield diffracted pressure is found, which includes a fluid-loaded, sub-sonic (relative to the water) flexural wave in the ice plate. Comparisons are also made with the locally reacting approximation to the input impedance of an ice plate. The combined effects of the ice lead diffraction process represent loss mechanisms that contribute to the transmission loss in long-range Arctic acoustic propagation.