Solution of the problem of interaction of Rayleigh waves with the edge of a wedgewith a small apex angle

Abstract

The problem of interaction of Rayleigh waves with the edge of a wedge with a small apex angle was solved using classical methods. This solution describes surface waves whose characteristics agree with well-known experimental data and bulk waves which have a significant effect on energy distribution. New features of surface wave propagation in wedge-shaped bodies are revealed.Introduction. Difficulties in the solution of the problem of interaction of Rayleigh waves with the edge of a wedge with a small apex angle are due to the complexity of the acoustic processes occurring near the edge of the wedge. The present work is a continuation of studies [1-3], which showed that in a Rayleigh wave, the directions of the wave normal of shear and longitudinal components do not coincide with the direction of wave propagation. This indicates acoustic anisotropy on the interface between the media and leads to unusual phenomena observed during reflection of the wave from the wedge faces.The wave field in a wedge is a set of four waves: the incident surface wave, the wave reflected from the edge of the wedge, the wave propagating to the second face of the wedge, and transformed bulk waves. The structure of Rayleigh waves is characterized by the presence of displacements localized near the surface. In approaching the edge, the displacements reach the opposite face of the wedge and cause perturbations of the this face, resulting in secondary waves. Because these perturbations are inhomogeneous, both surface and bulk waves arise which transfer energy in a direction opposite to the edge of the wedge. Therefore, in the case θ < 90 • , the parameters and spatial structure of the acoustic field incident on the edge change when the displacements reach the second face of the wedge.The evolution of the structure of the incident Rayleigh wave during motion from infinity to the edge of the wedge was investigated in [3], where an approximate solution of the wave problem considered was obtained using classical methods. To satisfy the condition of the absence of stresses on the faces of the wedge, the wave vector k should take complex values whose real part defines the wave velocity and whose imaginary part defines the decay due to energy outflow during transformation into bulk waves.Calculations show that, in approaching the edge, the wave is divided into two modes -antisymmetric and symmetric, whose spectral characteristics become similar to the characteristics of bulk waves (Fig. 1). As the wedge angle decreases, changes in the phase velocity are manifested in a certain increase in the steepness of the wave with increasing local thickness of the wedge at the location of this wave.We note that in elastic media, the propagation velocity of an ultrasonic wave remains constant, whereas in a wedge-shaped plate, it varies monotonically when approaching the edge of the wedge: the velocity of the symmetric mode increases and tends to the longitudinal wave velocity, and the velocity of the antisymmetric mode decreases to zero.

Stagnant Nonuniform Acoustic Wave

“…The cause of this phenomenon lies in the fact that the conditions of the surface wave propagation change: the displacements of particles in the media begin to reach the opposite face of the wedge that leads to the splitting of the wave to independent surface and spatial components [2][3]. The energy of the surface wave during the movement begins to decrease, since the spatial waves, continuously flowing from the surface into the medium, carry off some energy.…”