Propagation of pressure waves in the vicinity of a rigid inclusion submerged in a channel bounded by an elastic half-space
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1 Propagation of pressure waves in the vicinity of a rigid inclusion submerged in a channel bounded by an elastic half-space A. Tadeu, L. Godinho & J. Antonio Department of Civil Engineering University of Coimbra, Portugal Abstract The present paper computes the pressure field generated by spatially-sinusoidal harmonic pressure line sources in the vicinity of a rigid inclusion submerged in a fluid channel, bounded by an elastic half-space. The formulation of the problem is performed in the frequency domain. Time solutions are obtained by applying inverse Fourier transforms. The solution is computed using a D Boundary Element Method (BEM) model. Only the surface of the rigid inclusion is discretized with boundary elements, since the Green's functions used take the presence of the free surface and of the fluidlelastic interface floor into account. Computer simulations are performed for a fluid waveguide with a submerged rigid inclusion. The time domain responses are presented and discussed, identifying features that can be used to detect the presence and location of inclusions submerged in a fluid channel. The responses are compared with those provided by a model that assumes the floor of the fluid channel to be rigid, in order to assess the influence of the mechanical properties of the fluid floor channel on the fluid pressure variation. 1 Introduction In many practical problems, dynamic techniques based on the features of wave propagation in fluid media have been used to allow the detection of submerged objects. To allow a correct interpretation of these features, researchers have developed numerical and analytical models that can simulate underwater wave propagation under a variety of configurations.
2 Hackman et al. [l] and Sammelmann et a1.[2] describe a method based on the use of a transition matrix, and its application to the study of acoustic scattering inside inhomogeneous and homogeneous waveguides in the presence of scatterers. They developed solutions for the case of waveguides with planar geometry, built from a series of inhomogeneous fluid layers, expressing the acoustic wavefield in any layer as the sum of two terms: the first is the contribution from the source in terms of normal modes of the waveguide, while the second is the contribution from the scatterer, which takes the form of an extended source with unknown source strength coefficients. Specifically, they used this procedure to study the scattering from an elastic spherical shell inside a homogeneous fluid waveguide, bounded from above by a free surface and from below by a rigid surface. The scattering of acoustic waves by objects buried in underwater sediments was studied by Lim et al. [3], who presents both theoretical and experimental results. Their model was based on a full-wave transition matrix implementation of the Helmoltz equation, applied to the case of a layered fluid structure, with an upper fluid half-space containing the source, a lower fluid half-space containing an elastic scatterer, and an interfacial fluid layer between the two half-spaces. Using this model, they simulated the case of an elastic shell buried in the lower fluid and examined the specific consequences of burial for the free-field resonances. Numerical methods such as the Finite Element method or the Boundary Element Method (BEM) may also be used in the study of underwater sound propagation in the presence of submerged objects. The BEM is possibly the one best suited to analyze infinite or semi-infinite domains, since far field radiation conditions are automatically satisfied and only the discretization of interior boundaries is required. Dawson et al. [4] used a boundary integral equation method to compute the scattering of underwater sound from compact deformations of an oceanic waveguide surface. The method employs a Green's function that adequately satisfies the boundary conditions of the waveguide without the deformation, and that allows a sound speed variation with depth. The same method was used by Fawcett et al. [5,6] to compute the two-dimensional scattered field generated by objects embedded between two fluid half-spaces with different densities. Work by the authors of the present paper [7,8] has studied the pressure field generated by point sources placed inside a fluid channel with a rigid deformation on its floor. A BEM formulation in the frequency domain was used to model the floor deformation, and appropriate Green's functions, based on the superposition of virtual sources, were used to take into account the boundary conditions of the free surface, the rigid flat floor and the lateral walls confining the channel. In many of the works referred to above, the propagation domains are treated as fluid media, even when sedimentary layers are considered. This approach creates several limitations, since it neglects the shear rigidity of those materials. In other articles, where the bottom of the waveguide is assumed to be rigid, it is not possible to take into account the interaction between the fluid filling the waveguide and an elastic ground. In the present paper, a BEM formulation is used to study the scattering of pressure waves generated by a harmonic line pressure load inside a fluid
3 Boundary Elrmcnt Tcrhnology XV 367 waveguide with a free surface and elastic floor, simulating the presence of a sedimentary ground. Green's functions that take into account the required boundary conditions at the interface between the ground and fluid and at the free surface of the waveguide are used, avoiding the full discretization of these boundaries. These functions are analytical solutions for the steady state response of a homogeneous fluid layer, bounded from below by an elastic half-space and from above by a free surface, and subjected to a spatially sinusoidal harmonic line load. Results in the time domain are presented for a waveguide with a submerged rigid circular inclusion. Additionally, the results computed using this method are compared with those found when the bottom of the channel is assumed to be a rigid surface. 2 BEM formulation Figure 1 illustrates the geometry of the model, which assumes the existence of a submerged rigid inclusion inside a homogeneous fluid bounded by a flat elastic ground formation at the bottom (density p,, shear wave velocity P,, compressional wave velocity as ) and by a free surface on the top. The host fluid, with density,of, allows a compressional wave velocity af. This system is excited by a spatially sinusoidal harmonic pressure line load along the z direction, oscillating with a frequency w, located in the fluid at position (X,,, yo). Yt Figure 1: Schematic representation of the BEM model. The incident pressure field can be expressed by
4 in which, A is the wave amplitude, k, = Jo2/af2 - k with itn kaf 5 o, i = fi, k, is the axial wavenumber in the z direction and H!) (...) are second Hankel functions of order n. Assuming that this incident pressure wavefield strikes the boundary of the rigid inclusion, the following integral equation can be written, cp(xi,yk3kz.~) ~-~~(xk.yk>x~,~lrk~,a)~(x,>~l~ki~~)ds S (2) + pin' (xk yk kz,w) To obtain the solution of this integral equation for an arbitrary boundary surface (S), the surface is discretized into N straight boundary elements, for which boundary values pk are unknown. For constant pressure value boundary elements, Eq. (l l) takes the form, with H k1 = y,, X,, y,, k,, U) ds, where k is the loaded e~~ment, P' is the Cl pressure at the nodal point of element 1 and H(x,, y,, X,, y,, k,, U) is the pressure velocity Green's function at the nodal point of element l due to a pressure load at k. The factor c is a constant defined by the shape of the boundary, receiving the value 112 for a nodal point over a straight boundary segment. In the present work, the BEM is implemented with Green's functions that take into account the presence of an elastic ground and of a free surface. These functions are obtained as the summation of the field generated by a unit pressure load and the fields originated at the top and bottom of the waveguide. The components originated at the bottom and at the top surfaces may be expressed by the so-called surface terms, which, added to the incident field, verify the required boundary conditions: continuity of normal displacements and stresses, and null tangential stresses at the interface between the fluid and the elastic half-space; null pressures at the free surface. These Green's functions are defined making use of the solid displacement and fluid pressure potentials derived by Tadeu et al. [9, 101 in the definition of the Green's functions for a harmonic (steady state) line load, with a sinusoidally varying amplitude in the third dimension, placed in an infinite medium. The derivation of these Green's functions is omitted here for the sake of brevity. To validate these Green's solutions, the results computed for the geometric configuration displayed in Figure 2a were compared with those given by a BEM model that requires the discretization of the ground-fluid interface and the free surface, using Green's functions for an unbounded space. The mechanical properties of the elastic and fluid media are given in the same figure.
5 Boundary Elrmcnt Tcrhnology XV 369 A spatially sinusoidal harmonic pressure line load (k, = l.0radlm) is applied at X = O.0m and y = 2.0m. The calculations are performed in the frequency domain [ 1.O, 128.OHz ] with a frequency increment of 1.OH2. The imaginary part of the frequency has been set to = 0.7(2n/~), with T = 0.1 s. Figure 2b displays the real and imaginary parts of the scattered pressure field recorded at the receivers placed at X = 8.0m and y = 3.0m. The solid lines correspond to the solutions computed using the present Green's function, while the marked points represent the standard BEM solution. The round marks indicate the real part of the response, while the square marks refer to the imaginary part. The results show that there is an excellent agreement between the two solutions. Frequency (Hz) b) Figure 2: a) Geometric configuration; b) Frequency response at receiver RI. 3 Time responses After obtaining frequency domain responses, the pressure in the spatial-temporal domain is computed by a numerical fast inverse Fourier transform in U. For this purpose, the pressure point source is assumed to have a temporal variation defined by a Ricker pulse.
6 This technique allows the analysis of a total time window of T = 2z/Aw, where Aw is the frequency step. Pulses arriving at times later than Twill appear again in the beginning of this window, generating the so-called aliasing phenomenon. To prevent these pulses contributing to the result, complex frequencies of the form U, = U-ig (with 7 = 0.7Aw) are used (e.g. Phinney [l l]). In the time domain, this shift is taken into account by applying an exponential window e'" to the response (Kausel [12]). 4 Numerical simulations In the examples presented, a fluid waveguide 20.0m deep and filled with water (p, = 1000~~/rn~, ai = 1500m/s ) is assumed to be illuminated by a harmonic pressure line load placed 0.5m from its bottom. Two different physical models are simulated, corresponding to a waveguide with a rigid floor and to a waveguide limited from below by an elastic half-space ( p, = 1590 Q/m3, a, = l643 m/s, p, =526m/s ). Inside the waveguides, a circular cylindrical rigid inclusion of radius R is modeled using constant boundary elements, whose lengths are at least 12 times smaller than the wavelength of the incident pressure waves. A minimum of 20 elements is used. A line pressure load placed 0.5m away from the ground illuminates these systems, and pressure variations are computed at two lines of receivers, one placed 0.5m and the other 28.5m away from the source. A schematic representation of this geometry is presented in Figure 3. O+m 14.0m Om Receivers line 1 pf=l 000kglm3 at=l500rn/s Receivers line 2 Figure 3: Geometry of the model. For all the cases presented, the response is computed for a full set of frequencies ranging from 10.OHz to 1280.OHz, with an increment of 10.OHz. Time responses are then computed by applying an inverse Fourier transform and assuming that the excitation source generates a Ricker pulse with a characteristic frequency of 450.OHz.
7 Boundary Elrmcnt Tcrhnology XV 37 1 The first set of results, presented in Figure 4, refers to the case of a waveguide without the submerged object (R=O.Om). Figure 4a depicts the time responses computed at the second line of receivers, when the floor of the waveguide is assumed to be rigid. A grayscale is used to represent the amplitude of waves arriving at the receivers, with lighter shades corresponding to higher (positive) values, and darker shades representing negative values. The first signal registered at the receivers corresponds to the incident pulse, generated at the source point and travelling directly to the receivers. For the lower placed receiver ( from the ground), this pulse arrives at t = 28-~500.0 = 19.0ms, arriving at successively later times at receivers placed further away from the ground surface. After this first pulse, there follows a sequence of signals originated from multiple reflections on the top and bottom boundaries of the waveguide. When these reflections occur at the free surface, a 180" phase shift is apparent, which was expected due to the null pressure boundary conditions ascribed to that surface. Analyzing the results computed for a waveguide bounded by an elastic halfspace, shown in Figure 4b, there is a clear change in the response registered at the same line of receivers. As for the first case, the arrival of a first pulse travelling directly from the source to the receivers is clearly visible, but now with a lower amplitude. In fact, for the first case, there is a constructive combination between the incident pulse and a first reflection on the rigid floor, enhancing the response registered at the line of receivers. This phenomenon is less noticeable when the bottom of the waveguide is elastic. Since the ground floor allows a dilatational wave propagation velocity close to the pressure wave velocity of the fluid, a very significant part of the energy that hits the lower boundary is transmitted to this medium, and only the remaining part is reflected back to the fluid. The transmitted energy then propagates in the sediments as dilatational, shear and guided waves along the solid-fluid interface. It thus becomes evident that each time a pressure wave hits the fluid-sediment interface, it loses energy to the elastic medium, as can be clearly seen in Figure 4b. Another important feature of the response registered at this line of receivers is the presence of an additional pulse (labeled G), which is only visible at receivers placed near the fluidsediment interface. This pulse corresponds to a wave that is generated at the surface of the elastic medium, designated as a guided wave, traveling with velocities approaching 447.0ds. The velocity of these waves is close to that of the Rayleigh waves. When a rigid inclusion is submerged in the fluid waveguide, the response suffers significant changes, as can be seen in the time domain responses in Figure 5, computed for the two waveguides described before. Figure 5a illustrates the behavior registered at the two lines of receivers when the floor of the waveguide is modeled as a rigid surface and for an inclusion defined by R=l.Om. As seen before, the first signal arriving at the receivers corresponds to the incident pulse, traveling directly from the source. As expected, the arrival of this pulse at line 1 occurs at the beginning of the time window, while it arrives much later at line 2. Again, the pulse generated by the source hits the top and bottom surfaces of the waveguide, suffering successive reflections that arrive at both lines of receivers
8 and forming a regular pattern. However, the presence of other pulses is now evident, and can be clearly distinguished in both plots, These are caused by the reflection and diffraction of waves on the-rigid surface of the submerged object which then suffer multiple reflections on the rigid and free surfaces of the waveguide, appearing in the time responses as a second regular pattern with a significant but lower amplitude. As an example, for lower placed receivers in line l, the arrival time of the first reflection of this type can be determined as t = 26-x500.0 = 17.7ms. After the first arrivals coming from the rigid object, another pulse is visible, resulting from a pulse reflected on the ground after being first reflected by the rigid scatterer. Higher order reflections also become visible at the two lines of receivers, forming successive patterns of lower amplitude. They correspond to further interactions that occur between the submerged object and the surfaces of the waveguide. a) b) Figure 4: Time response registered at line 2 receivers when R=O.Om: a) waveguide iith rigidfloor; b) waveguide with elastic floor. This scenario changes to some extent when a sediment ground is considered. Time domain responses obtained for this model at the same lines of receivers are displayed in Figures 5b. The features described in the absence of any inclusion can still be identified in the time domain plots. Again, the first pulse arriving at the receivers corresponds to the directly incident pulse, followed by a sequence of reflections on the two surfaces of the waveguide that arrive with successively lower amplitudes. Other patterns appear superimposed in the time response, due to the presence of the rigid inclusion. The arrival times coincide with those found for the waveguide with rigid floor, although there is a significant decrease of amplitude as energy is transmitted to the sediment half-space. Higher order reflections on all the surfaces involved (top, bottom and rigid inclusion) occur, and a sequence of pulses arrives at successively later times. One should note that the responses registered at the first and second lines of receivers differ substantially, both for the waveguide with rigid floor and for the case of the sediment half-space. In fact, for the first line, the dominant phenomenon involved is the reflection of waves on the different surfaces. For this case, the pattern of pulses that is generated by the reflections of the incident wave
9 Boundary Elrmcnt Tcrhnology XV 373 on the top and bottom surfaces is clearly separated and easily distinguishable from the reflections generated at the rigid inclusion. On the other hand, the response at line 2 is strongly marked by the diffraction of waves on the inclusion and the different types of pulses arriving at the receivers cannot be so clearly identified. Receivers Line 1 Receivers Line tirnefrns) tirneirne) b) Figure 5: Time response registered when R=l.Om: a) waveguide with rigid floor; b) waveguide with elastic floor. S Final Remarks This paper has presented a numerical study of the pressure field generated by spatially-sinusoidal harmonic pressure line sources in the vicinity of a rigid inclusion submerged in a fluid channel. Two different cases were studied, corresponding to the case of a waveguide with a rigid floor and to a waveguide bounded from below by a sediment half-space. A frequency domain BEM formulation was used, in conjunction with appropriate Green's functions, to take into account the presence of the rigid floor or of the sediment half-space. Time solutions are subsequently computed by means of an inverse Fourier transform. The proposed implementation was found to be efficient, allowing the successful study of pressure wave propagation inside fluid-filled waveguides with different bottom surface boundary conditions.
10 The findings described here show that there are substantial differences between the two models compared. The assumption of a rigid floor at the bottom of the waveguide leads to wavefields with higher amplitudes, which may not occur in a real model. By contrast, when the bottom of the waveguide is an elastic material, the energy transmission to the underlying medium is evident. The detection of a circular inclusion submerged in the fluid could be performed for both cases analyzed. However, for the rigid model, the amplitude of the reflected and diffracted field coming from the scattering object shows very high amplitudes. It is also possible to conclude, for both cases, that detection is much easier and more reliable if the vertical line of receivers is placed near the pressure source, since the pulses originated by wave reflections on the surface of the rigid inclusion are clearly separated. References Hackman, R., Sammelmann G., Acoustic Scattering in an lnhomogeneous Waveguide: Theory, J. Acoust. Soc. Am., Vol 80(5), pp , Sammelmann, G., Hackrnan, R., "Acoustic Scattering in an Homogeneous Waveguide, J. Acoust. Soc. Am., Vol82(1), pp , Lim, R., Lopes, J., Hackman, R., Todoroff, D., Acoustics scattering in an inhomogeneous waveguide, J. Acoust. Soc. Am., Vol 93(4), pp , Dawson, T., Fawcett, J., A boundary integral equation method for acoustic scattering in a waveguide with nonplanar surfaces, J Acousl. Soc. Am, Vol87, pp , Fawcett, J., The computation of the scattered pressure field from a cylinder embedded between two half-spaces with different densities, J Acoust. Soc. Am., Vol99, pp , Fawcett, J., Acoustic scattering fi-om cylindrical objects embedded between two half-spaces, J. Acoust. boc. Am., Vol 100, pp , Godinho, L., Tadeu, A., Branco, F., 3D Acoustic Scattering from an Irregular Fluid Waveguide via the BEM, EABE - Engineering Analysis with Boundary Elements Journal, Vol25(6), pp ,2001. Branco, F.; Godinho, L.; Tadeu, A., Propagation of Pressure Waves Inside a Confined Fluid Channel with an Irregular Floor, Journal of Computationul Acoustics, Vol 10(2), pp ,2002. Tadeu, A., Antonio, J., 2.5D Green's functions for elastodynamic problems in layered acoustic and elastic formations", J Computer Modeling in Eng and Sciences,Vol. 2(4), pp , [l01 Tadeu, A., Kausel, E., Green's functions for two-and-a-half dimensional elastodynamic problems", Jouvnal of Engineering Mechanics, ASCE, Vol 126(10), pp ,2000. [l l] Phinney, R. A., Theoretical calculation of the spectrum of hst arrivals in layered elastic mediums, J. Geophys. Res., 70, pp , [l21 Kausel, E., Roesset, J. M., Frequency domain analysis of undamped systems, Journal of Engineering Mechanics, ASCE, Vol. 1 18, No. 4, pp , 1992.
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