Luís Godinho

Analytical evaluation of the acoustic insulation provided by double infinite walls

Abstract The acoustic insulation provided by infinite double panel walls, when subjected to spatially sinusoidal line pressure loads, is computed analytically. The methodology used extends earlier work by the authors on the definition of the acoustic insulation conferred by a single panel wall. It does not entail any simplification other than the assumption that the panels are of infinite extent. The full interaction between the fluid (air) and the solid layers is thus taken into account and the calculation does not involve limiting the thickness of any layer, as the Kirchhoff or Mindlin theories require. The problem is first formulated in the frequency domain. Time domain solutions are then obtained by means of inverse Fourier transforms using complex frequencies. The model is first used to compute the sound reduction provided by a double homogeneous brick wall, with identical panels, when illuminated by plane sound waves. The results are then compared with those provided by the simplified method proposed by London, which was later extended by Beranek (London–Beranek method). The limitations of the simplified London–Beranek model, namely, its applicability only to double walls with identical mass, subjected to plane waves, and its failure to account for the coincidence effect, are overcome by the method proposed. Time signatures are produced to illustrate the different sound transmission mechanisms. Several types of body and guided waves are originated, giving rise to a complex dynamic system with multiple reflections within the solid and fluid layers and the global resonance of the system. The effect of the cavity absorption is considered by attributing a complex density to the air filling the space between the two wall panels. Absorption attenuates the dips of insulation controlled by the cavity resonances. Several simulations are then performed for different combinations of wall and air layer thickness to assess the influence of this variable on the final acoustic insulation. The influence of the air cavity on sound reduction was found to be dependent on the frequency. At low frequencies a better performance was achieved for thicker air layers, while at higher frequencies a thinner air layer is preferable. The use of wall panels with different mass resulted in the wall performing better, particularly for high frequencies.

3D sound scattering by rigid barriers in the vicinity of tall buildings

The boundary element method (BEM) is used to evaluate the acoustic scattering of a threedimensional (3D) sound source by an infinitely long rigid barrier in the vicinity of tall buildings. The barrier is assumed to be non-absorbing and the buildings are modeled as an infinite barrier. The calculations are performed in the frequency domain and time signatures are obtained by means of inverse Fourier transforms. The 3D solution is obtained by means of Fourier transform in the direction in which the geometry does not vary. This requires solving a series of 2D problems with different spatial wavenumbers, kz. The wavenumber transform in discrete form is obtained by considering an infinite number of virtual point sources equally spaced along the z axis. Complex frequencies are used to minimize the influence of these neighboring fictitious sources. Different geometric models, with barriers of varying sizes, are used. The reduction of sound pressure in the vicinity of the buildings is evaluated and the creation of shadow zones by the barriers is analyzed and compared with results provided by a simplified method. # 2001 Elsevier Science Ltd. All rights reserved.

Numerical Simulation of Ground Rotations along 2D Topographical Profiles under the Incidence of Elastic Plane Waves

The surface displacement field along a topographical profile of an elastic half-space subjected to the incidence of elastic waves can be computed using different numerical methods. The method of fundamental solutions (MFS) is one of such tech- niques in which the diffracted field is constructed by means of a representation in terms of the Greens functions for discrete forces located outside the domain of inter- est. From the enforcement of boundary conditions, such forces can be computed; thus, the ground motion can be calculated. One important advantage of MFS over boundary integral techniques is that singularities are avoided. The computation of ground- motion rotations implies the application of the rotational operator to the displacement field. This can be done using either numerical derivatives or analytical expressions to compute the rotational Greens tensor. We validate the method using exact analytical solutions in terms of both displacement and rotation, which are known for simple geometries. To demonstrate the accuracy for generic geometries, we compare results against those obtained using the spectral-element method. We compute surface rota- tions for incoming plane waves (P, SV, and Rayleigh) near a topographical profile. We point out the effects of topography on rotational ground motion in both frequency and time domains. Online Material: Analysis of the dependence of rotational motion on incident plane-wave frequency.

Wave motion between two fluid-filled boreholes in an elastic medium

The boundary element method (BEM) is used to fully simulate the propagation of waves between two fluid-filled boreholes. The sources are placed in one of the boreholes while the receivers are placed in the other. This model is frequently used in cross-hole seismic prospecting techniques to assess the characteristics of the elastic medium between the two boreholes. This work studies the dependence of the wave propagation patterns on the distance between the source and the receiver, their location and orientation relative to the axis of a circular borehole and type of elastic formation (fast and slow formations). In addition, this BEM model is used to compute the influence of the deformed boreholes whose cross-section is not circular. Both the spectra responses and the time-domain responses are computed to elucidate the main physical features of the problem solved.

Performance of the BEM solution in 3D acoustic wave scattering

Abstract A fixed cylindrical circular cavity and a cylindrical circular column of fluid of infinite length submerged in a homogeneous fluid medium, and subjected to a pressure point source, for which closed form solutions are known, are used to assess the performance of constant, linear and quadratic boundary elements in the analysis of acoustic scattering. This aim is accomplished by evaluating the error committed by the boundary element method (BEM) for a wide range of frequencies and wave numbers. First, the position of dominant BEM errors in the frequency versus spatial wave number domains are identified and related to the natural modes of vibration of the cylindrical circular inclusion. Then, the errors that occur by using constant, linear and quadratic elements are compared when the inclusion is modelled with the same number of nodes (i.e. maintaining computational cost). Finally, the importance of the position of the nodal points inside discontinuous boundary elements is analysed.

Sound propagation around rigid barriers laterally confined by tall buildings

Abstract This paper focuses on the propagation of sound waves in the presence of acoustic barriers placed close to very tall buildings. The boundary element method (BEM) is used to model the acoustic barrier, while the presence of the tall buildings is taken into account by using the image source method. Different geometries are analyzed, representing the cases of a single building, two buildings forming a corner and three buildings defining a laterally confined space. The acoustic barrier is assumed to be non-absorbing, and all the buildings and the ground are modeled as infinite rigid plane surfaces. Calculations are performed in the frequency domain and time signals are then obtained by means of Inverse Fourier Transforms. The sound pressure loss provided by the acoustic barrier is computed, illustrating the importance of the lateral confinements.

Scattering of acoustic waves by movable lightweight elastic screens

Abstract This paper computes the insertion loss provided by movable lightweight elastic screens, placed over an elastic half-space, when subjected to spatially sinusoidal harmonic line pressure sources. A gap between the acoustic screen and the elastic floor is allowed. The problem is formulated in the frequency domain via the boundary element method (BEM). The Greens functions used in the BEM formulation permit the solution to be obtained without the discretization of the flat solid–ground interface. Thus, only the boundary of the elastic screen is modeled, which allows the BEM to be efficient even for high frequencies of excitation. The formulation of the problem takes into account the full interaction between the fluid (air) and the solid elastic interfaces. The validation of the algorithm uses a BEM model, which incorporates the Greens functions for a full space, requiring the full discretization of the ground. The model developed is then used to simulate the wave propagation in the vicinity of lightweight elastic screens with different dimensions and geometries. Both frequency and insertion loss results are computed over a grid of receivers. These results are also compared with those obtained with a rigid barrier and an infinite elastic panel.

3D acoustic scattering from an irregular fluid waveguide via the BEM

The BEM is used to calculate the variation in the pressure field generated by a dilatational point load inside a channel filled with a homogeneous fluid, in the presence of an irregular floor. The Greens functions are defined in the frequency domain and obtained by superposing virtual acoustic sources combined so as to generate the boundary conditions of the free or rigid surfaces of the channel. The responses in the time domain are obtained by means of Fourier transforms, making use of complex frequencies. The main features and spectral representation of the signals scattered by irregular floors are then described and used to elucidate the most important aspect of wave acoustics, which can provide the basis for the development of non-destructive testing and imaging methods.

Wave scattering by infinite cylindrical shell structures submerged in a fluid medium

Abstract This work analyzes the wave scattering by an elastic, fluid-filled, cylindrical shell structure submerged in a fluid medium and subjected to the effect of a point pressure load placed inside or outside the cylindrical shell. The shell structures modeled have constant cross-sections along their axis, corresponding to 2.5D problems. A Fourier transformation in the direction in which the geometry does not vary is applied to find the 3D field as the summation of the 2D solutions for different spatial wavenumbers. The wave propagation patterns are analyzed for shell structures defined by two concentric or non-concentric cylindrical circular surfaces. The boundary element method, formulated in the frequency domain, is used to calculate the dynamic response of these systems when the shell is defined by non-concentric cylindrical circular surfaces, while analytical solutions are used to compute the response when two concentric cylindrical circular surfaces define the shell structure. Different simulations are performed for receivers placed at distinct positions, in order to study the normal modes excited at each load position within each shell structure type. Both frequency and time solutions are obtained, in an attempt to obtain wave propagation features that may be used as a basis for developing non-destructive testing techniques.

Three-dimensional wave scattering by a fixed cylindrical inclusion submerged in a fluid medium

Abstract This paper presents the solution for a fixed cylindrical irregular cavity of infinite length submerged in a homogeneous fluid medium, and subjected to dilatational point sources placed at some point in the fluid. The solution is first computed for a wide range of frequencies and wavenumbers, which are then used to obtain time-series by means of (fast) inverse Fourier transforms into space–time. The method and the expressions presented are implemented and validated by applying them to a fixed cylindrical circular cavity submerged in an infinite homogeneous fluid medium subjected to a point pressure source for which the solution is calculated in closed form. The boundary elements method is then used to evaluate the wave-field elicited by a point pressure source in the presence of fixed rigid cylindrical cavities, with different cross-sections, submerged in an unbounded fluid medium and in a half-space. Simulation analyses with this idealized model are then used to study the patterns of wave propagation in the vicinity of these inclusions. The amplitude of the wavefield in the frequency vs axial-wavenumber domain is presented, allowing the recognition, identification, and physical interpretation of the variation of the wavefield.