Modeling linear Rayleigh wave sound fields generated by angle beam wedge transducers

Abstract

In this study, the reciprocity theorem for elastodynamics is transformed into integral representations, and the fundamental solutions of wave motion equations are obtained using Green’s function method that yields the integral expressions of sound beams of both bulk and Rayleigh waves. In addition to this, a novel surface integral expression for propagating Rayleigh waves generated by angle beam wedge transducers along the surface is developed. Simulation results show that the magnitudes of Rayleigh wave displacements predicted by this model are not dependent on the frequencies and sizes of transducers. Moreover, they are more numerically stable than those obtained by the 3-D Rayleigh wave model. This model is also applicable to calculation of Rayleigh wave beams under the wedge when sound sources are assumed to radiate waves in the forward direction. Because the proposed model takes into account the actual calculated sound sources under the wedge, it can be applied to Rayleigh wave transducers with different wedge geometries. This work provides an effective and general tool to calculate linear Rayleigh sound fields generated by angle beam wedge transducers.

On the Directivity of Acoustic Waves Generated by the Angle Beam Wedge Actuator in Thin-Walled Structures

“…In NDE and SHM practice, the angle beam wedge actuators, which use the principles of refraction and mode conversion to produce refracted shear or longitudinal waves in the test material, are very efficient for the inspection of thick metallic components as they provide a large scanning index. These abilities have been studied and confirmed by earlier [23][24][25][26] and later theoretical and experimental research [16,27,28]. The first work dedicated to the use of the wedge actuator for the wave generation in thin-walled plates was [29], where the wedge method of generating guided waves was analyzed with particular attention being focused on the relationship between the angularly dependent excitation amplitude of a given mode and the physical parameters of the transducer and wedge used to excite the mode.…”

“…The first work dedicated to the use of the wedge actuator for the wave generation in thin-walled plates was [29], where the wedge method of generating guided waves was analyzed with particular attention being focused on the relationship between the angularly dependent excitation amplitude of a given mode and the physical parameters of the transducer and wedge used to excite the mode. This theoretical investigation used the analytical approach assuming that the transducer produces a roughly parabolic pressure distribution on the contact footprint of the form where σ 0 represents the maximum pressure which occurs at the center of the transducer face, α = 0, and the transducer has a width D. More detailed later studies [16,[26][27][28] based on the finite-element analysis established a very complex contact stress distribution that produces both out-of-plane and in-plane displacements within the elliptic footprint. Later papers [14,15] discuss difficulties of the wave tuning at its excitation in thin-walled structures.…”

Measurement of Rayleigh Wave Beams Using Angle Beam Wedge Transducers as the Transmitter and Receiver with Consideration of Beam Spreading

“…The workaround for solving this problem is that we apply the reciprocity theorem to develop the integral representation theorem, derive fundamental solution of the wave motion equations using Green’s function method, and apply the solution into the actual condition [6]. …”

“…Considering all the contributions from the actual sound source positions, the area sound source can be replaced by these line sources and the line integral formula from Equation (18) can be extended to the following form to express the surface integral as [6]:${\mathrm{u}}_{\mathrm{r}}{false(\mathrm{normalx}}_2{,\mathrm{normaly}}_{2}false)=\frac{-2}{{\mathrm{c}}_{\mathrm{r}}{\mathrm{L}}_{\mathrm{x}}}true{\int }_{{\mathrm{S}}_{\mathrm{F}}}{\mathrm{normalv}}_1^{\prime }{(\mathrm{x}}_2{,\mathrm{y}}_2{)\mathrm{G}(\mathrm{x}}_2{,\mathrm{y}}_2|{\mathrm{normalx}}_2^{\prime },{\mathrm{normaly}}_2^{\prime }{)\mathrm{dS}}_{\mathrm{normalF}}$ where ${\mathrm{L}}_{\mathrm{x}}$ is the equivalent length of the footprint area source, which is introduced to calculate the sound beams out of the footprint and given as:${\mathrm{L}}_{\mathrm{x}}={2\mathrm{normala}/\cos \mathrm{sans-serif\theta }}_{\mathrm{normalp}1}$ where $\mathrm{a}$ is the radius of the contact transducer. The integral area should cover the whole area underneath the wedge, although the sound sources out of the footprint have little effect on the Rayleigh wave beam distributions.…”

“…We should note that this model equation is also valid for this case. When the wedge partially covers the footprint, the actual sound source should be used, and the new equivalent length ${\mathrm{L}}_{\mathrm{x}}^{\prime }$ should also be selected [6]. …”