On energy balance and the structure of radiated waves in kinetics of crystalline defects

Traveling waves, with well-known closed form expressions, in the context of the defects kinetics in crystals are excavated further with respect to their inherent structure of oscillatory components. These are associated with, so called, Frenkel–Kontorova model with a piecewise quadratic substrate potential, corresponding to the symmetric as well as asymmetric energy wells of the substrate, displacive phase transitions in bistable chains, and brittle fracture in triangular lattice strips under mode III conditions. The paper demonstrates that the power expended theorem holds so that the sum of rate of working and the rate of total energy flux into a control strip moving steadily with the defect equals the rate of energy sinking into the defect, in the sense of N.F. Mott. In the conservative case of the Frenkel–Kontorova model with asymmetric energy wells, this leads to an alternative expression for the mobility in terms of the energy flux through radiated lattice waves. An application of the same to the case of martensitic phase boundary and a crack, propagating uniformly in bistable chains and triangular lattice strips, respectively, is also provided and the energy release is expressed in terms of the radiated energy flux directly. The equivalence between the well-known expressions and their alternative is established via an elementary identity, which is stated and proved in the paper as the zero lemma. An intimate connection between the three distinct types of defects is, thus, revealed in the framework of energy balance, via a structural similarity between the corresponding variants of the ‘zero’ lemma containing the information about radiated energy flux. An extension to the dissipative models, in the presence of linear viscous damping, is detailed and analog of the zero lemma is proved. The analysis is relevant to the dynamics of dislocations, brittle cracks, and martensitic phase boundaries, besides possible applications to analogous physical contexts which are marked by macroscopic energy release through emission of waves and possibly linear viscous damping.     

Rayleigh waves obtained by the indeterminate couple-stress theory

Rayleigh waves in a linear elastic couple-stress medium are investigated; the constitutive equations involve a length parameter l that characterizes the microstructure of the material. With , c_T=conventional transversal speed and q=wave number, an explicit expression is derived for the relation between , lq and Poisson's ratio ν. The Rayleigh speed turns out to be dispersive and always larger than the conventional Rayleigh speed. It is of interest that when lq=1 and ν≥0, it always holds that . The displacement field is investigated and it is shown that no Rayleigh wave motions exist when lq→∞ and when lq=1, ν≥0. Moreover, a principal change of the displacement field occurs when lq passes unity. The peculiarity that no Rayleigh wave motions exist when lq=1, ν≥0 may support the criticism by Eringen (1968) against the couple-stress theory adopted here as well as in much recent literature.     

Fully nonlinear modeling of radiated waves generated by floating flared structures

The nonlinear radiated waves generated by a structure in forced motion, are simulated numerically based on the potential theory. A fully nonlinear numerical model is developed by using a higher-order boundary element method （HOBEM）. In this model, the instantaneous body position and the transient free surface are updated at each time step. A Lagrangian technique is employed as the time marching scheme on the free surface. The mesh regridding and interpolation methods are adopted to deal with the possible numerical instability. Several auxiliary functions are proposed to calculate the wave loads indirectly, instead of directly predicting the temporal derivative of the velocity potential. Numerical experiments are carried out to simulate the heave motions of a submerged sphere in infinite water depth, the heave and pitch motions of a truncated flared cylinder in finite depth. The results are verified against the published numerical results to ensure the effectiveness of the proposed model. Moreover, a series of higher harmonic waves and force components are obtained by the Fourier transformation to investigate the nonlinear effect of oscillation frequency. The difference among fully nonlinear, body-nonlinear and linear results is analyzed. It is found that the nonlinearity due to free surface and body surface has significant influences on the numerical results of the radiated waves and forces.     

Movement of the ground in Rayleigh waves produced by underground explosions

Analytic representations are obtained for the displacement and stress fields in the Rayleigh surface wave (R-wave) generated in an elastic half-space by an internal source that produces the same seismic P-wave as an underground explosion. Oscillograms, particle trajectories, and stresses in the half-space and on its surface are calculated. Relations for the energy flux in the R-wave are obtained. For rock salt, the fraction of the explosion energy transferred to the R-wave is estimated. It is established that this fraction can reach values of about 1% of the total explosion energy if the explosion is a contained one. As the charge depth is increased, the energy of the R-wave decreases in approximately inverse proportion to the depth. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 47, No. 4, pp. 3–14, July–August, 2006.     

On the energy transfer equation for weakly interacting waves

The derivation of the transfer equation based on analysis of the equations for spectral semi-invariant and not invoking equations for realization of the random wave field is presented. Uniformly valid asymptotic expansions for the third and the fourth spectral semi-invariant are constructed using the multiple scale method and the matched asymptotic expansion method. This approach makes it possible to investigate the boundary layer in a neighbourhood of the resonant surface where intensive growth in time of the third spectral semi-invariant occurs. This boundary layer defines the form of the transfer equations. An analogous boundary layer for the fourth spectral semiinvariant and its influence on the second and the third spectral semi-invariants are also investigated.     

Scattering of pulsed Rayleigh surface waves by a cylindrical cavity

A pulsed Rayleigh surface wave of prescribed shape is incident on a cylindrical cavity which is parallel to both the plane free surface and the plane wave front. Multiple reflections at the cylindrical and plane free surface are considered and the resulting displacements and stress components are calculated in the surrounding of the cavity by approximately summing infinite double sums. Use is made of the stationary loading case simulated by a periodic train of wave pulses and its time Fourier series representation and of expansions of all incident and reflected waves in terms of cylindrical wave functions. For reflection, the free surface of the half-space is approximated by a fictitious convex (or concave) cylindrical surface of “large” radius. The wave pattern due to a single pulse loading is constructed from the stationary solution by enforcing homogeneous initial conditions in the half-space ahead of the single loading pulse and by prescribing a wide spacing in the periodically set-forth train of pulses. The numerical results for stresses and dynamic stress magnification factors are especially useful for the interpretation of recent measurements in dynamic photoelasticity.     

On the propagation of energy in linear conservative waves

Summary This paper is concerned with the question when and why the rate of energy propagation in a system of waves equals the group velocity. It is shown by the method of stationary phase that this equality holds, for travelling waves without dissipation, whenever this method applies. The reason why this result can be obtained by this kinematical method is investigated by a discussion of simple harmonic waves. It is shown that the choice of an expression for the energy density to be used in connection with a given wave equation is restricted by the conservation of energy in such a way that the average rate of work done divided by the average energy density always equals the group velocity. Finally some examples of wave motion are discussed to illustrate the derived formulae.     

On Rayleigh scattering by a grating

The acoustic diffraction of a plane wave by a periodic row of identical cylinders of arbitrary cross section and characteristic dimension a is calculated for ka ? kd < π, where k is the wave number and d is the wavelength of the array. The reflection and transmission coefficients depend only on d, k, the angle of incidence, and the area and virtual mass of the cross section. The general results are applied to a grating of inclined flat plates.     

Scattering of Rayleigh and Stoneley waves by two adhering elastic wedges

The method of Sommerfeld integrals is used to study propagation of Rayleigh and Stoneley waves in a system of two bonded solid wedges with a common vertex. Numerical results are obtained for configurations with a wide range of angles of steel and aluminum wedges.     

Rayleigh waves in orthotropic fluid-saturated porous media

In this paper, we are interested in the propagation of Rayleigh waves in orthotropic fluid-saturated porous media. This problem was investigated by Liu and Liu (2004). The authors have derived the secular equation of the wave but that secular equation is still in implicit form. The main aim of this paper is to derive explicit secular equation of the wave. By employing the method of polarization vector, the secular equations of Rayleigh waves in explicit form is obtained. This equation recovers the dispersion equation of Rayleigh waves propagating in pure orthotropic elastic half-spaces. Remarkably, the secular equation obtained is not a complex equation as the one derived by Liu and Liu, it is a really real equation.     

On aerodynamic noises radiated by the pantograph system of high-speed trains

Pantograph system of high-speed trains become significant source of aerodynamic noise when travelling speed exceeds 300 km/h. In this paper, a hybrid method of non-linear acoustic solver （NLAS） and Ffowcs Williams-Hawkings （FW-H） acoustic analogy is used to predict the aerodynamic noise of pantograph system in this speed range. When the simulation method is validated by a benchmark problem of flows around a cylinder of finite span, we calculate the near flow field and far acoustic field surrounding the pantograph system. And then, the frequency spectra and acoustic attenuation with distance are analyzed, showing that the pantograph system noise is a typical broadband one with most acoustic power restricted in the medium-high frequency range from 200 Hz to 5 kHz. The aerodynamic noise of pantograph systems radiates outwards in the form of spherical waves in the far field. Analysis of the overall sound pressure level （OASPL） at different speeds exhibits that the acoustic power grows approximately as the 4th power of train speed. The comparison of noise reduction effects for four types of pantograph covers demonstrates that only case 1 can lessen the total noise by about 3 dB as baffles on both sides can shield sound wave in the spanwise direction. The covers produce additional aerodynamic noise themselves in the other three cases and lead to the rise of OASPLs.     

Propagation of acoustic waves and shock waves radiated by a sinusoidal pulsation of a sphere during a single period

A spherical sound wave is emitted by a sphere which executes a small sinusoidal pulsation of a single period at high frequency in an inviscid fluid. Nonlinear propagation of the waves is formulated as an initial boundary value problem and is analysed in detail. The governing equation is linear near the sphere, while it is a nonlinear hyperbolic equation in a far field. The nonlinearity has a significant effect there, leading to the formation of two shocks. The exact solution to match the near field solution can easily be obtained for the far field equation. The nonlinear distortion of waveform and the shock formation distance are evaluated from the representation of the solution with strained coordinates. The evolution and nonlinear attenuation of the two shock discontinuities are also examined by making use of the equal-areas rule. In its asymptotic form the entire profile is an N wave with a long tail.     

Propagation of Rayleigh waves on curved surfaces

The propagation and properties of Rayleigh waves on curved surfaces are investigated theoretically. The Rayleigh wave dispersion equation for propagation on a curved surface is derived as a parabolic equation, and its penetration depth is analyzed using the curved surface boundary. Reciprocity is introduced to model the diffracted Rayleigh wave beams. Simulations of Rayleigh waves on some canonical curved surfaces are carried out, and the results are used to quantify the influence of curvature. It is found that the velocity of the surface wave increases with greater concave surface curvature, and a Rayleigh wave no longer exists once the surface wave velocity exceeds the bulk shear wave velocity. Moreover, the predicted wave penetration depth indicates that the energy in the Rayleigh wave is transferred to other modes and cannot propagate on convex surfaces with large curvature. A strong directional dependence is observed for the propagation of Rayleigh waves in different directions on surfaces with complex curvatures. Thus, it is important to include dispersion effects when considering Rayleigh wave propagation on curved surfaces.     

Computational studies of inertia-gravity waves radiated from upper tropospheric jets

The generation and physical characteristics of inertia-gravity waves radiated from an unstable forced jet at the tropopause are investigated through high-resolution numerical simulations of the three-dimensional Navier–Stokes anelastic equations. Such waves are induced by Kelvin–Helmholtz instabilities on the flanks of the inhomogeneously stratified jet. From the evolution of the averaged momentum flux above the jet, it is found that gravity waves are continuously radiated after the shear-stratified flow reaches a quasi-equilibrium state. The time–vertical coordinate cross-sections of potential temperature show phase patterns indicating upward energy propagation. The sign of the momentum flux above and below the jet further confirms this, indicating that the group velocity of the generated waves is pointing away from the jet core region. Space–time spectral analysis at the upper flank level of the jet shows a broad spectral band, with different phase speeds. The spectra obtained in the stratosphere above the jet show a shift toward lower frequencies and larger spatial scales compared to the spectra found in the jet region. The three-dimensional character of the generated waves is confirmed by analysis of the co-spectra of the spanwise and vertical velocities. Imposing the background rotation modifies the polarization relation between the horizontal wind components. This out-of-phase relation is evidenced by the hodograph of the horizontal wind vector, further confirming the upward energy propagation. The background rotation also causes the co-spectra of the waves high above the jet core to be asymmetric in the spanwise modes, with contributions from modes with negative wavenumbers dominating the co-spectra. Dedicated to the memory of our colleague Dr. Binson Joseph     

Rayleigh waves of arbitrary profile in anisotropic media

The paper deals with surface wave propagation in an orthorhombic elastic half-plane. The general profile of the wave is considered, incorporating the anisotropy effects within the known representation in terms of a single plane harmonic function.     

Growing crack as superposition of surface Rayleigh waves

A crack, symmetrically propagating in elastic material, was considered as superposition of surface Rayleigh waves. The self-similar growth of face loaded crack was considered in detail. Exact expressions of deformation and stress fields in the crack’s surrounding were found and asymptotic behavior of stress near crack’s tips was also obtained. A condition that determines the crack’s velocity of self-similar propagation was found.     

Rayleigh lamb waves in micropolar isotropic elastic plate

The propagation of waves in a homogeneous isotropic micropolar elastic cylindrical plate subjected to stress free conditions is investigated. The secular equations for symmetric and skew symmetric wave mode propagation are derived. At short wave limit, the secular equations for symmetric and skew symmetric waves in a stress free circular plate reduces to Rayleigh surface wave frequency equation. Thin plate results are also obtained. The amplitudes of displacements and microrotation components are obtained and depicted graphically. Some special cases are also deduced from the present investigations. The secular equations for symmetric and skew symmetric modes are also presented graphically.     

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.     

Rayleigh waves emitted by a propagating crack in a strain-rate dependent elastic medium

A simple model was proposed for the interpretation of the non-circular form of the Rayleigh wavefronts emitted by a fast running crack in a plate. The surface deformation around the crack tip, due to the high stress concentration there, propagated as a surface wave after fracture of this zone. On the other hand, the moving singularity of the crack tip created a dynamic stress field of varying intensity with time all over the specimen. This dynamic stress field resulted in a significant change of the mechanical properties of a strain-rate dependent material and therefore it influenced the velocity of propagation of fracture-Rayleigh wavefronts. An analysis of this varying dynamic strain field explained the non-circular form of Rayleigh waves, accompanying the propagating crack. For the experimental evaluation of the K₁-factor the method of dynamic caustics was used in conjunction with the high-speed photography technique.