Diffraction of sound by an elastic cylinder near the surface of an elastic halfspace

Diffraction of an acoustic wave by an elastic cylinder near the surface of an elastic halfspace is considered. The solution relies on a Helmholtz-type integral equation and uses the Green function of an elastic halfspace. The latter function is represented in the form of an integral over the Sommerfeld contour on the plane of a complex variable that has the meaning of the angle of the wave incidence on the halfspace boundary. An integral equation for the sound pressure distribution over the cylinder surface is derived. This equation is reduced to an infinite system of equations for the Fourier-series expansion coefficients of this distribution. The results obtained are valid for the diffraction of a cylindrical wave and a plane wave. They also describe the diffraction of a spherical wave when the transmitter and receiver are far from the cylinder and lie in one plane that is orthogonal to the cylinder axis.     

Acoustic analysis of a rectangular cavity with general impedance boundary conditions

A Fourier series method is proposed for the acoustic analysis of a rectangular cavity with impedance boundary conditions arbitrarily specified on any of the walls. The sound pressure is expressed as the combination of a three-dimensional Fourier cosine series and six supplementary two-dimensional expansions introduced to ensure (accelerate) the uniform and absolute convergence (rate) of the series representation in the cavity including the boundary surfaces. The expansion coefficients are determined using the Rayleigh-Ritz method. Since the pressure field is constructed adequately smooth throughout the entire solution domain, the Rayleigh-Ritz solution is mathematically equivalent to what is obtained from a strong formulation based on directly solving the governing equations and the boundary conditions. To unify the treatments of arbitrary nonuniform impedance boundary conditions, the impedance distribution function on each specified surface is invariantly expressed as a double Fourier series expansion so that all the relevant integrals can be calculated analytically. The modal parameters for the acoustic cavity can be simultaneously obtained from solving a standard matrix eigenvalue problem instead of iteratively solving a nonlinear transcendental equation as in the existing methods. Several numerical examples are presented to demonstrate the effectiveness and reliability of the current method for various impedance boundary conditions, including nonuniform impedance distributions.     

Directional properties of an array of sound receivers positioned in an impedance screen recess

Expressions for calculating the directional characteristics of an array of sound receivers positioned in a waveguide with impedance walls are obtained from the solution to the problem on the diffraction of a plane sound wave by the waveguide open end with impedance flanges. The waveguide can be of a finite length, and, in this case, it can be considered as an open cavity in an impedance screen. The solution of the integral equation for the sound pressure distribution over the opening area is reduced to the solution of an infinite system of algebraic equations for the coefficients of the field expansion in normal waveguide waves. Examples of calculated directional characteristics are presented for arrays with receivers positioned at different distances from the opening and for different values of the impedances of the waveguide walls and flanges.     

The diffraction of sound by an impedance sphere in the vicinity of a ground surface

The problem of sound diffraction by an absorbing sphere due to a monopole point source was investigated. The theoretical models were extended to consider the case of sound diffraction by an absorbing sphere with a locally reacting boundary or an extended reaction boundary placed above an outdoor ground surface of finite impedance. The separation of variables techniques and appropriate wave field expansions were used to derive the analytical solutions. By adopting an image method, the solutions could be formulated to account for the multiple scattering of sound between the sphere and its image near a flat acoustically hard or an impedance ground. The effect of ground on the reflected sound fields was incorporated in the theoretical model by employing an approximate analytical solution known as the Weyl-van der Pol formula. An approximation solution was suggested to determine the scattering coefficients from a set of linearly coupled complex equations for an absorbing sphere not too close to the ground. The approximate method substantially reduced the computational time for calculating the sound field. Preliminary measurements were conducted to characterize the acoustical properties of an absorbing sphere made of open cell polyurethane foam. Subsequent experiments were carried out to demonstrate the validity of the proposed theoretical models for various source/receiver configurations around the sphere above an acoustically hard ground and an impedance ground. Satisfactory comparative results were obtained between the theoretical predictions and experimental data. It was found that the theoretical predictions derived from the approximate solution agreed well with the results obtained by using the exact solutions.     

Piston-type radiator in a soft screen in the pekeris waveguide

A model problem is considered for a radiator in the form of a circular disk with a given pressure jump at its surface. The radiator is inserted in a soft screen coinciding with the upper boundary of the Pekeris waveguide. A series expansion of the sound field in normal modes is obtained. A numerical analysis of the radiation impedance and its components that are responsible for the radiation into the waveguide and into the halfspace is carried out.     

Excitation of seismoacoustic waves by a harmonic force source acting at the boundary of a liquid layer and an elastic halfspace

A problem on the excitation of seismoacoustic waves in a system of a homogeneous isotropic elastic halfspace covered with a liquid layer is solved in the case of action of a source of point harmonic force on the surface of an elastic medium. Integral expressions are obtained for the radiation powers averaged over a wave period for longitudinal and transverse waves in a solid. Mode excitation is analyzed in detail. Expressions describing parts of the mode powers radiated into a liquid layer and an elastic medium are obtained. Numerical analysis of radiation powers is conducted for spherical longitudinal and transverse waves as well as for the radiation powers of seismoacoustic modes in a solid halfspace and a liquid layer. It is determined that in the conditions characteristic of bottom rocks in the case, where the basin depth is several times and more larger than the sound’s wavelength, about 2/3 of the total power is radiated into a liquid.     

A vertical eigenfunction expansion for the propagation of sound in a downward-refracting atmosphere over a complex impedance plane

The propagation of sound in a stratified downward-refracting atmosphere over a complex impedance plane is studied. The problem is solved by separating the wave equation into vertical and horizontal parts. The vertical part has non-self-adjoint boundary conditions, so that the well-known expansion in orthonormal eigenfunctions cannot be used. Instead, a less widely known eigenfunction expansion for non-self-adjoint ordinary differential operators is employed. As in the self-adjoint case, this expansion separates the acoustic field into a ducted part, expressed as a sum over modes which decrease exponentially with height, and an upwardly propagating part, expressed as an integral over modes which are asymptotically (with height) plane waves. The eigenvalues associated with the modes in this eigenfunction expansion are, in general, complex valued. A technique is introduced which expresses the non-self-adjoint problem as a perturbation of a self-adjoint one, allowing one to efficiently find the complex eigenvalues without having to resort to searches in the complex plane. Finally, an application is made to a model for the nighttime boundary layer.     

On the sound field of a shallow spherical shell in an infinite baffle

A method is presented for calculating the far field sound radiation from a shallow spherical shell in an acoustic medium. The shell has a concentrated ring mass boundary condition at its perimeter representing a loudspeaker voice coil and is excited by a concentrated ring force exerted by the end of the voice coil. A Green's function is developed for a shallow spherical shell, which is based upon Reissner's solution to the shell wave equation [Q. Appl. Math. 13, 279-290 (1955)]. The shell is then coupled to the surrounding acoustic medium using an eigenfunction expansion, with unknown coefficients, for its deflection. The resulting surface pressure distribution is solved using the King integral together with the free space Green's function in cylindrical coordinates. In order to eliminate the need for numerical integration, the radiation (coupling) integrals are solved analytically to yield fast converging expansions. Hence, a set of simultaneous equations is obtained which is solved for the coefficients of the eigenfunction expansion. These coefficients are finally used in formulas for the far field sound radiation.     

Green's functions and Brewster condition for a halfspace bounded by an anisotropic impedance plane

Dyadic Green's functions are obtained for a halfspace bounded by an anisotropic impedance plane. Using the Fresnel reflection coefficients, these functions are derived in planewave spectral forms. The Brewster condition is also obtained.     

Diffraction of point-source-generated sound by an elastic cylindrical shell

The three-dimensional problem of the scattering of a harmonic sound wave by an elastic cylindrical shell is solved using Debye potentials. All potentials are represented in the form of integrals depending on the axial component of the wave vector.     

Sound propagation along an impedance plane

The sound field due to a point source above a plane boundary with a constant normal impedance is obtained by a double saddle point method of integration. Variations in previous studies by Ingard, by Lawhead and Rudnick and by Wenzel are clarified.     

Modified theory of the physical-optics approach to the impedance wedge problem

The problem of a wedge with equal face impedances is examined with a modified theory of physical optics. The surface integral is constructed by use of the impedance boundary condition. The aperture equivalent current is estimated from the behavior of the reflected diffracted field. The integrals obtained are evaluated asymptotically and compared with the exact solution numerically.     

Vibrations of an elastic strip with a rough boundary

A plane problem of steady-state forced vibrations of an elastic strip whose lower boundary contains a rough segment is considered. Using Green’s functions for a strip, the problem is reduced to a system of integral equations with integrals over the rough boundary, which is solved by the boundary-element method. The inverse problem of determining the shape of the rough boundary segment from the data on the displacement field of a certain part of the upper boundary is formulated. By the linearization procedure, the inverse problem is reduced to a Fredholm integral equation of the first kind with a smooth kernel, which is solved by Tikhonov’s regularization method.     

Numerical modeling of the reflection coefficients for the plane sound wave reflection from a layered elastic bottom

The matrix method and its numerical realization are considered in calculating the complex reflection coefficients and refraction indices of plane sound waves for geoacoustic models of the ocean bottom in the form of homogeneous elastic (liquid) absorbing layers overlying an elastic halfspace. In calculating the reflection coefficients at high frequencies or in the presence of a large numbers of sedimentary layers, a passage from the Thomson-Haskell matrix approach to the Dunkin-Thrower computational scheme is performed. The results of test calculations are presented. With the aim of developing resonance methods for the reconstruction of the parameters of layered elastic media, the behavior of the frequency-angular dependences of the reflection coefficient are studied for various geoacoustic bottom models. The structure of the angular and frequency resonances of the reflection coefficients is revealed. The dependence of the structure (the position, width, and amplitude) of two types of resonances on the parameters of the layered bottom is considered.     

半空间球面波函数叠加法重构声源直接辐射声场

提出了基于半空间球面波函数叠加的声场重构方法,以重构含有限声阻抗边界半空间中声源直接辐射的声场。在半空间中多极子声源声压场的解析解的基础上,构造出以边界声阻抗为参量的半空间球面波函数的正交基;通过求逆获得半空间总声压解的基函数系数,同时也获得声源直接辐射声场即自由空间中的基函数系数,进而重构出声源直接辐射的声场。在边界声阻抗已知和边界声阻抗未知两种条件下,对该方法进行了仿真验证和参数分析,并在全消声室内进行了实验验证。结果表明,所提方法能重构出半空间中典型声源即球形声源和平面声源的直接辐射声场;该方法在边界声阻抗已知时的重构精度与稳定性高于在边界声阻抗未知时的情形。      

Diffraction of a sound wave by the open end of a flanged waveguide with impedance walls

Diffraction of a plane sound wave by the open end of an impedance-wall waveguide connected to an opening in an impedance screen is considered. The plane wave is incident on the waveguide from a free half-space. Two versions of the problem are considered: for a semi-infinite waveguide and for a finite-length waveguide with a specified bottom impedance; the impedances of the walls, screen, and waveguide bottom can be different. The finite-length waveguide can be treated as an open cavity in the impedance screen. For the cavity of zero length, the problem is reduced to the diffraction by an impedance insert in the impedance screen. The solution in the external region determines the scattered field; the solution in the internal region allows one to determine the directional pattern of an array of receivers located in the cavity. The problem is solved using the integral Helmholtz equation with a specially selected Green’s function that provides the fulfillment of the boundary conditions. Formally, the problem is reduced to an infinite system of algebraic equations. The computational results obtained for bistatic and monostatic scattering patterns are presented.     

Diffraction of a Plane Electromagnetic Wave by an Infinite Double-Ridge Wedge with a Dielectric Cylinder at the Crest

We reduce the considered problem to solving a matrix equation of the second kind for unknown coefficients of expansion of a diffracted field into a Fourier–Bessel series. This expansion was obtained by imposing boundary conditions on the diffracted field with the subsequent re-expansion of the field function over basis functions in a given interval. The expansion coefficients were determined analytically in the case where the electric diameter of the cylinder is less than unity as well as numerically with a high accuracy by solving the obtained matrix equation using the reduction method. We derived expressions for the pattern of the far-zone field scattered by the studied structure and the backscattering cross section and give exact numerical results for the case of an E-polarized incident wave.     

Reflection of plane waves from an elastic layered medium: Resonance approach and numerical modeling

The resonance formalism developed earlier by H. Überall for a model of a liquid layer overlying a liquid halfspace is extended to a model of an elastic layer overlying an elastic halfspace. Using the Thomson—Haskell matrix technique, an exact analytical expression is obtained for the complex reflection coefficient. Characteristic equations are derived, and their roots, which determine the positions of the resonances of the reflection coefficient for longitudinal and transverse waves, are obtained analytically. In the resonance approach, the exact expression for the reflection coefficient is replaced by an approximate one that describes the behavior of the reflection coefficient near the resonances. The comparison of the exact and approximate values of the reflection coefficient shows good agreement between the results near the frequency and angular resonances.     

Diffraction of sound from a dipole source near to a barrier or an impedance discontinuity

Pierce's formulation for the diffraction of spherical waves by a hard wedge has been extended to the case of the sound field due to a dipole source. The same approach is also used to extend a semiempirical model for sound propagation above an impedance discontinuity due to a dipole source. The resulting formulas have been validated by comparing their numerical solutions with that computed by summing the sound fields due to two closely spaced monopole sources of equal magnitude but opposite in phase. These new formulations are then used to develop a simple model for calculating the dipole sound field diffracted by a barrier above an impedance ground. Applications of these models relate to transportation noise prediction, particularly railway noise abatement, for which dipole sources are commonly used. The numerical predictions have been found to compare reasonably well with indoor measurements using piezoceramic transducers as dipole sources.     

Acoustic characteristics analysis of slender cavity system with arbitrary impedance end conditions

A modeling method for the dynamic characteristics analysis of a slender acoustical cavity with impedance end conditions is established. In order to satisfy the continuity requirement at impedance ends for the first order differential of sound pressure, field function is constructed as the standard Fourier series supplemented by boundary smoothed auxiliary polynomials. System characteristic equation is derived by solving the governing differential equation and impedance acoustic boundary of slender acoustical cavity system simultaneously,relevant acoustical modal information is obtained via the state space solution procedure. In numerical simulation, various acoustic variables, such as acoustical modal frequency, sound pressure modal shape, sound pressure response and the particle velocity, are presented for the slender acoustical cavity system with different boundary conditions and compared with those results in the existing literature. The correctness and effectiveness of the proposed method are then fully validated.