Reflection operator methods for elastic waves II — composite regions and source problems

Reflection and transmission operator methods are particularly effective in problems where multiple reflections are important. When two regions are joined together, the overall reflection operators for the composite region can be built up from the reflection and transmission operators for the individual regions. The two regions are linked by a sequence of successive reflections between the zones which may be represented via a reverberation operator. For source problems, the model can be broken along a surface passing through the source and the overall displacement can be found by following the physical sequence of propagation processes. This leads to a representation in terms of reflection and transmission operators for the regions above and below the source. The two zones are linked by the reverberation operator for the structure, which determines the modal characteristics in the model.     

The integral operator representation of propagation invariants for elastic waves in irregularly layered media

The reflection and transmission of elastic waves in irregularly layered media can be represented in terms of a set of propagation invariants for elastic wavefields. There are a number of such invariants depending on different integral operators acting on the displacement and traction fields. By exploiting different classes of invariants, representations for reflection and transmission operators at an irregular interface can be derived which are easier to compute than previous forms.

In a similar way, a simplified representation for the surface amplification factor for incident waves on a free surface can be constructed. This is particularly useful for calculating theoretical seismograms on the surface of a body.     

Reflection and transmission of obliquely incident Rayleigh surface waves by a viscoelastic interphase

The reflection and transmission of obliquely incident Rayleight surface waves by an interphase between two quarter spaces of identical or different materials, have been investigated. The mechanical behavior of the interphase is represented by a thin viscoelastic layer. By using the full space Green's functions due to a spatially harmonic line load, the mathematical statement of the 3-dimension problem is reduced to a 2-dimension system of singular integral equations. The far-field behavior of the scattered waves leads to the definition of reflection and transmission coefficients,R andT. The system of the singular integral equations are solved forR andT with the boundary element method. The results are presented for selected values of the elastic constants of the joined quarter spaces, the parameters of the interphase and the incident angles of Rayleigh surface waves.     

The boundary integral equation method for the solution of wave-obstacle interaction

The boundary integral equation method constitutes the basis of a number of computer programs used for the solution of wave-obstacle interaction problems. For the case of obstacles in a constant depth fluid, the method assumes that the velocity potential at any point in the fluid may be represented by a distribution of Green's function sources over the immersed surface of the obstacle. Application of the obstacle kinematic boundary condition gives rise to an integral equation which may be solved, using numerical discretization, for the unknown source strength distribution function. Subsequent evaluation of the discretized velocity potential permits evaluation of the hydrodynamic interaction parameters. A series of numerical solutions have been carried out for a range of substantially rectangular obstacles, in a two-dimensional domain, using varying levels of immersed profile discretization. The results, presented in the form of fixed and floating mode wave reflection and transmission, together with the motion response of the floating obstacle, demonstrate the significant sensitivity of the evaluated parameters to variations in the level of discretization.     

Theory of generalized rays for SH-waves in dipping layers

The theory of generalized rays is applied to analyze transient waves in a layered half-space with non-parallel interfaces. The propagation, transmission, reflection, and refraction of SH waves which are generated by a line source in the surface layer of a three-layer model are considered, each of the two overlaying layers having a different dipping angle.Generalized ray integrals for multi-reflected rays in the top layer and for rays that are transmitted into the lower layer and then refracted back into the top layer are formulated by using three rotated coordinate systems, one for each interface, and are expressed in terms of local wave slowness along each interface. Through a series of transformations of the local slownesses, all ray integrals are expressible in a common slowness variable. The arrival time of each ray undergoing multiple reflections and transmissions is then determined from the stationary value of the phase function with common slowness of the ray integral. Inverse Laplace transform of these ray integrals are completed by Cagniard's method.     

Scattering and excitation of SH-surface waves by a protrusion at the mass-loaded boundary of an elastic half-space

A rigorous theory of the scattering and excitation of SH-surface waves by a protrusion at the mass-loaded boundary of an elastic half-space is presented. The boundary value problem (which is of the third kind) is solved by employing two suitably chosen Green functions. One of them is represented as a Fourier type of integral, the other is taken to be the Bessel function of the second kind and order zero. The procedure leads to a system of three, coupled, integral equations. This system is solved numerically. In case of an incident bulk wave, the amplitude of the launched surface wave is computed; in case of an incident surface wave, its transmission and reflection factor are computed. For both cases, an expression for the far-field radiation pattern of the scattered bulk wave is derived. A reciprocity relation is shown to exist between the amplitude of the launched surface wave and the far-field bulk wave radiation pattern. Numerical results are presented for a triangularly-prismatic protrusion; they are compared with the results pertaining to a corresponding indentation in the mass-loaded boundary, that have been obtained in a previous paper.     

Reflection and transmission of antiplane surface waves by a surface-breaking crack in a layered elastic solid

The reflection and transmission of antiplane surface waves (Love waves) by a surface-breaking crack in a layered elastic solid is investigated. The crack is normal to the free surface, and breaks into the lower half-space solid. The formulation of the problem is reduced to a singular integral equation of the Cauchy type. In this equation, the unknown function, which is the slope of the crack-face displacement, is discontinuous at the interface between the two solids. It is shown that the magnitude of the discontinuity is related to the ratio of the shear moduli. A Gaussian numerical method is used to obtain the solution of the singular integral equation. At some distance from the plane of the crack, the wave motion is the superposition of a finite number of Love-wave modes. The amplitudes of these modes are readily evaluated in terms of the slope of the crack-face displacement. Curves are presented for the reflection coefficients corresponding to the first three modes and for the transmission coefficient as functions of the dimensionless frequency.     

Scattering of scalar waves from a rough interface using a single integral equation

Using Green's function methods we consider the problem of scattering from a rough interface separating two semi-infinite homogenous media. We derive a single coordinate-space integral equation of the first kind for the generalized reflection coefficient R. A second integral equation of the first kind is derived for the generalized transmission coefficient T. The two equations are new results. In the limiting cases corresponding to Dirichlet and Neumann boundary conditions, we recover the usual boundary integral equations from the R-equation. In the flat surface limit, R and T reduce to the usual Fresnel reflection and transmission coefficients. The latter are derived from these more general results rather than assumed as in perturbation methods.     

On the application of branched operator continued fractions for a boundary problem of linear viscoelasticity

In solving linear viscoelastic problems for composite materials, the problem arises of representing a multivariable operator function. To resolve this problem, the method of operator continued fractions is generalized to the case of a multivariable operator function. The method is based on the theory of branched continued fractions. Branched operator continued fractions are considered. Using the convolution theorem, fractions can be represented in terms of operators of basic class. This representation makes it possible to effectively solve boundary problems of linear viscoelasticity Published in Prikladnaya Mekhanika, Vol. 42, No. 1, pp. 133–143, January 2006.     

Invariant imbedding method for wave problems

A new method, which reduces various boundary value problems for a wave equation to initial value problems, is developed. The scalar Helmholtz equation for the one- and three-dimensional cases is considered. The method is extended to the case of nonlinear media. Its applications to the wave equation for different dimensions and various media are described. The source of the wave field may be situated outside or inside the layer occupied by the medium. Governing equations are obtained for cases when one can neglect the backward scattering. The operators that arise are reduced to integral operators. The problem of wave scattering by a weakly rough surface is briefly considered.     

Classification of pseudo-steady shock wave reflection types

Classification of various types of the reflections of a shock wave over a straight wedge is proposed. The idea about entire reflection phenomenon as a result of interaction of two processes—the shock wave reflection process and the flow deflection process—serves as a basis for the classification. To recognize the types of reflection, changes in the shapes of the reflected wave, Mach stem, and contact surface (slipstream) are taken into account. The boundaries and domains of existence for various types of reflection configuration are reported. New terms for some types of reflection are proposed. The domain of irregular non-Mach reflection is analyzed carefully. It is shown that the von Neumann reflection pattern can result from not only the weak shock reflection but also the strong shock reflection over thin wedges. Shadowgraph images of different types of irregular reflection that illustrate the suggested classification are presented. Emphasis is placed on near-wall behavior of the contact discontinuity in the Mach configuration.     

A green's function approach to the determination of internal fields

A time-domain technique is presented for computing the internal electromagnetic field within a one-dimensional medium characterized by spatially varying conductivity and permittivity profiles. A Green's operator is defined which maps the incident fields on either side of the medium to the field at an arbitrary observation point. This operator is shown to be a matrix of integral operators with kernels satisfying known partial differential equations and various other initial and boundary identities. A scheme for numerically calculating these kernels is presented along with a few examples of the calculations. Moreover, consideration of the boundary values of the Green's operator reveals a novel way for computing the reflection and transmission operators for the medium. Finally, a Green's function approach to internal fields in the presence of a phase velocity mismatch at one of the boundaries of the medium is outlined in an appendix.     

Diffuse holographic interferometric observation of shock wave reflection from a skewed wedge

The pattern of shock wave reflection over a wedge is, in general, either a regular reflection or a Mach reflection, depending on wedge angles, shock wave Mach numbers, and specific heat ratios of gases. However, regular and Mach reflections can coexist, in particular, over a three-dimensional wedge surface, whose inclination angles locally vary normal to the direction of shock propagation. This paper reports a result of diffuse double exposure holographic interferometric observations of shock wave reflections over a skewed wedge surface placed in a 100 × 180 mm shock tube. The wedge consists of a straight generating line whose local inclination angle varies continuously from 30° to 60°. Painting its surface with fluorescent spray paint and irradiating its surface with a collimated object beam at a time interval of a few microseconds, we succeeded in visualizing three-dimensional shock reflection over the skewed wedge surface. Experiments were performed at shock Mach numbers, 1.55, 2.02, and 2.53 in air. From reconstructed holographic images, we estimated critical transition angles at these shock wave Mach numbers and found that these were very close to those over straight wedges. This is attributable to the flow three-dimensionality.      

Sommerfeld-type integrals for discrete diffraction problems

Three problems for a discrete analog of the Helmholtz equation are studied analytically using the plane wave decomposition and the Sommerfeld integral approach. They are: (1) the problem with a point source on an entire plane; (2) the problem of diffraction by a Dirichlet half-line; (3) the problem of diffraction by a Dirichlet right angle. It is shown that the total field can be represented as an integral of an algebraic function over a contour drawn on some manifold. The latter is a torus. As a result, explicit solutions are obtained in terms of recursive relations (for the Green’s function), algebraic functions (for the half-line problem), or elliptic functions (for the right angle problem).     

Diffraction by a double grating

The reflection and transmission of a plane electromagnetic wave by a double grating is investigated. The double grating consists of two mutually parallel, planar arrays of perfectly conducting strips of vanishing thickness. Two types of polarization are investigated, viz.E- and H-polarization. For both types, integral equations for the unknown current densities in the two strips belonging to a single period of the grating, are derived. Subsequently, these integral equations are solved numerically, whereupon the reflection and transmission factors for the different spectral orders are computed.     

Study of the Deformation of Anisotropic Viscoelastic Bodies

A study is made of methods for solving linear viscoelastic problems on the basis of the Volterra concept — representation of irrational functions of integral operators as operator power series (analogues of Taylor series). It is pointed out that these series converge weakly. The results of development and substantiation of a new mathematical method for solution of the above problems are summarized. It is based on representing irrational functions of integral operators by operator continued fractions, which converge well. Solutions to certain linear viscoelastic problems for anisotropic bodies are given     

Scattering of a Rayleigh wave by an elastic three-quarter space

The steady state problem of the scattering of an incident Rayleigh wave by an elastic three-quarter space (270° wedge) is considered. The problem is reduced to the numerical solution of a pair of Fredholm integral equations of the second kind. The kernels of these equations consist of elementary functions. The reflection and transmission coefficients can be computed from two independent equations, which provides a check on both the accuracy and correctness of the results. The method used can be extended to other wedge angles.     

基于电磁声激励Lamb波的裂纹深度检测

本文采用电磁声传感器接收单一S₀模态激励,测量Lamb波在板中不同深度的槽形裂纹处的反射与透射,用于对已知扩展长度的表面裂纹进行深度测量。然后分析了Lamb波模态的频散与波动特性,由超声Lamb波的波结构,近似计算了S₀模态入射到裂纹时的反射系数。计算结果与实验结果一致。结果表明,Lamb波在有限长裂纹处的反射系数可采用二维模型中的反射系数求解公式近似求解。采用电磁超声传感器接收单一S₀模态激励,重复测量稳定性高,适用于Lamb波反射系数的测量。Lamb波的反射系数与裂纹深度有很好的对应关系,可用于板表面已知长度裂纹的深度测量。     

Energy identities in water wave theory for free-surface boundary condition with higher-order derivatives

A modified form of Green's integral theorem is employed to derive the energy identity in any water wave diffraction problem in a single-layer fluid for free-surface boundary condition with higher-order derivatives. For a two-layer fluid with free-surface boundary condition involving higher-order derivatives, two forms of energy identities involving transmission and reflection coefficients for any wave diffraction problem are also derived here by the same method. Based on this modified Green's theorem, hydrodynamic relations such as the energy-conservation principle and modified Haskind-Hanaoka relation are derived for radiation and diffraction problems in a single as well as two-layer fluid.