Rayleigh Critical Angle Research Articles

Evaluation of material damage using non-specular reflection of a bounded ultrasonic beam at a liquid-solid interface

This paper proposes an effective method for evaluating the level of material damage using the non-specular reflection of a bounded ultrasonic beam at a liquid-solid interface. Non-specular reflection occurs when a bounded ultrasonic beam is obliquely incident from a liquid onto a solid surface. This paper formally formulates the acoustic field of the non-specular reflection of a bounded ultrasonic beam and analyzes the influences of some factors such as the incident angle and the level of material damage (characterized by a decrease in Young's modulus) on the non-specular reflection field. To characterize the effect of the non-specular reflection of a bounded ultrasonic beam, this paper defines a specific parameter called the specular reflection coefficient (SRC) of a bounded ultrasonic beam. A finite-element (FE) model is established to quantitatively observe the response of the defined SRC to the incident angle and the level of material damage. The FE simulations are conducted for the bounded ultrasonic beam obliquely incident at longitudinal, transverse, and Rayleigh critical angles, respectively. The simulations demonstrate that the change rate of SRC at the Rayleigh critical angle exhibits a monotonically sensitive response to the level of material damage, providing an effective means for accurately evaluating it. To validate this method, we conducted an experiment with oblique incidence testing of the bounded ultrasonic beam at the Rayleigh critical angle of the water-aluminum interface. The experimental results are consistent with the FE simulations, and show that the change rate of SRC at the Rayleigh critical angle has a monotonous and sensitive response to the level of material damage. The consistency of theoretical predictions, FE simulations, and experimental results confirms the feasibility of using the non-specular reflection of a bounded ultrasonic beam incident at the Rayleigh critical angle as an effective method for evaluating the level of material damage.

An algorithm for quantitatively modeling reflected ultrasonic bounded pulses and beams

The study of the reflected acoustic waves plays an important role in our understanding of media. We provide an algorithm to propagate the ultrasonic bounded beam source and study its reflection from any horizontal and homogenous water-solid boundary. This algorithm implements a hybrid combination of the phase-advance wavefield continuation in the frequency domain and the complex analytic solution for the acoustic reflectivity. The peak amplitude of the specularly reflected beam is in agreement with the laboratory measured acoustic reflection from water-Aluminum and water-Copper alloy boundaries. The algorithm is able to model the observed critical reflection as well as the null in the reflected amplitude at the Rayleigh critical angle from the acoustic wave. This algorithm is a crucial tool to understand the full reflected wave from material immersed in water in any azimuthal or incidental angles. The software of this algorithm and acoustic reflectivity from both solid materials are provided.

Quantitative modeling of reflected ultrasonic bounded beams and a new estimate of the Schoch shift

The wavefields of bounded acoustic beams and pulses reflected from water-loaded plates are fully modeled with the phase advance technique. The wavefield produced at the source is propagated at any incidence angle using phase shift modeling that incorporates the full analytic solution for the acoustic reflectivity at the interface. This approach provides for the ready visualization of both the stationary monofrequency beam wavefield and animation of the temporally bounded pulse. The model images are reminiscent of the classic Schlieren photographs that first illustrated the nonspecular behavior of the reflected beams incident near critical angles. Various phenomena such as the lateral displacement and the null zone at the Rayleigh critical angle are recreated. A new approximation for this shift agrees well with that of the peak energy of the reflected beam. Similar effects are observed during the reflection of a bounded pulse. Although more computationally costly than existing analytic approximations, the phase advance technique can facilitate the interpretation of reflectivity measurements obtained in laboratory experiments. In particular, the full visualization allows for a better understanding of the behavior of reflected waves at any angle of incidence.

Acoustic reflectivity goniometry of bounded ultrasonic pulses: Experimental verification of numerical models

The angle of incidence reflectivity of a bounded acoustic pulse from water loaded elastic solids is tested using an experimental configuration with a large transmitter and a small pointlike receiver. The experimental response, affected by well-known nonspecular effects near critical angles, is modeled fully by propagation of the full bounded pulse using a phase advance technique. This allows for visualization of the evolution of the bounded pulse with time and upon reflection. The modeled and the observed reflectivities and phase rotations, as measured along the specular path, agree well with observations even at post Rayleigh critical angles of incidence. Moreover, small amplitude deviations are accurately reproduced. The longitudinal and Rayleigh critical angles are obtained from the observed reflectivity curves and used to calculate the material elastic wave speeds, which are in good agreement with those measured directly. The density is also obtained by inversion of portions of the reflectivity curves where they do not significantly differ from the plane wave response. These tests validate use of the full phase propagation method to study such reflection problems.

DPSM technique for ultrasonic field modelling near fluid–solid interface

Distributed point source method (DPSM) is gradually gaining popularity in the field of non-destructive evaluation (NDE). DPSM is a semi-analytical technique that can be used to calculate the ultrasonic fields produced by transducers of finite dimension placed in homogeneous or non-homogeneous media. This technique has been already used to model ultrasonic fields in homogeneous and multi-layered fluid structures. In this paper the method is extended to model the ultrasonic fields generated in both fluid and solid media near a fluid–solid interface when the transducer is placed in the fluid half-space near the interface. Most results in this paper are generated by the newly developed DPSM technique that requires matrix inversion. This technique is identified as the matrix inversion based DPSM technique. Some of these results are compared with the results produced by the Rayleigh–Sommerfield integral based DPSM technique. Theory behind both matrix inversion based and Rayleigh–Sommerfield integral based DPSM techniques is presented in this paper. The matrix inversion based DPSM technique is found to be very efficient for computing the ultrasonic field in non-homogeneous materials. One objective of this study is to model ultrasonic fields in both solids and fluids generated by the leaky Rayleigh wave when finite size transducers are inclined at Rayleigh critical angles. This phenomenon has been correctly modelled by the technique. It should be mentioned here that techniques based on paraxial assumptions fail to model the critical reflection phenomenon. Other advantages of the DPSM technique compared to the currently available techniques for transducer radiation modelling are discussed in the paper under Introduction.

Experimental Study of Non‐Specular Reflected Acoustic Field an a Water‐FSPM Interface

AbstractIn this paper, the non‐specular acoustic reflected field of pulsed beam incident on a Water‐FSPM (fluid saturated porus media, abbreviated as FSPM) interface was studied experimentally by the optical method and the transmitter‐receiver method. The reflection coefficient R of the sample in the range of the incident angle equal to 16°—80° is measured. The reflected angle of two beams, their beam displacements, their wave forms and spectra are recorded in the range of the incident angle equal to ±4° near the Rayleigh critical angle θc. During the experiments, in the range of θi equal to ±4° near θc two reflected beams A and B are observed. The measured displacement Δ of the beam B decreases as the incident angle θi increases and is not the maximum at θi = θc. The spectrum of the beam with the reflected angle which is equal to θi is much close to that of the incident beam, and the spectrum of the other beam with the reflected angle equal to θc has a negative shift to that of the incident beam. The phase of beam A is the same as that of directly received from transmitter, and the phase of the beam B is inverse to that of beam A.

Scattering of an obliquely incident acoustic wave by an infinite hollow cylindrical shell

Acoustic scattering from an isotropic elastic hollow cylindrical shell of infinite length excited by an obliquely incident plane acoustic wave is investigated. The form functions of an aluminum cylindrical shell immersed in water have been calculated by the direct summation of the Rayleigh series. Computations are made at angles (with the normal to the cylinder axis) between α=0° and α=35°. The results of the theoretical calculation are in good agreement with the results of experiments. The experimental results have shown in a frequency range of k1a=0 –20 that the resonances are related to three wave families: the circumferential wave (l=2) detected for angles smaller than the ‘‘angle of longitudinal wave in thin rods’’ (αl), the guided wave (p=1) detected for angles smaller than the second critical angle (αT), and the Scholte–Stoneley wave (l=0). The evolution of the resonance frequencies is followed for different angles and one can note experimentally, that at an angle superior to the Rayleigh critical angle (α=30.3°), resonances of the Scholte–Stoneley wave have been observed.

Determination of Sound Velocities by High Frequency Complex V(z) Measurement in Acoustic Microscopy

An amplitude-and-phase-measuring acoustic microscope operating at frequencies higher than 100 MHz has been developed. Complex V(z) curves of fused silica and aluminum were measured at 400 MHz with high accuracy. Inverse Fourier transform of each V(z) curve was achieved to give the reflectance function of the specimens. The Rayleigh, longitudinal and shear critical angles were obtained from the results. These were in good agreement with the theoretical predictions.

Use of a Conical Axicon as a Surface Acoustic Wave Focusing Device

Ultrasonic axicons generate waves which focus on a line. They are used in various imaging applications as hulk wave focusing devices with a very long depth of focus. A new type of conical axicon is introduced. It consists of a concave parabolic surface immersed in a liquid medium and insonified obliquely by wavefronts generated by a plane transducer. The parabolic cylinder can be approximated by a portion of a circular cylinder without losing significantly in the focus- ing performance of the axicon. It is also shown that conical axicons can be used to excite surface waves provided that the cone angle of the axicon coincides with the Rayleigh critical angle of the liquid-solid in- terface. The generated surface waves focus into a diffraction-limited spot. This new surface wave focusing scheme is easy to use and has a conversion efficiency and sensitivity far better than other existing tech- niques.

Focusing surface waves using an axicon

Axicons are generators of waves which focus on a line. They are used in various imaging and nondestructive testing applications as bulk wave focusing devices with a very long depth of focus. In this letter, a new type of conical axicon is introduced and it is shown that this axicon, immersed in a liquid, insonifying a plane solid surface can be used to excite surface waves on the solid surface provided that the cone angle of the axicon coincides with the Rayleigh critical angle of the liquid-solid interface. The generated surface waves focus into a diffraction-limited spot. This new surface wave focusing scheme is easy to use, has a conversion efficiency and sensitivity far better than other existing techniques.

Back reflection of ultrasonic waves from a liquid-solid interface

The problem of back reflection of ultrasound from a liquid-solid interface is considered analytically. In experiments it has been observed that the back reflected energy radiates from the interface at the Rayleigh critical angle, irrespective of the incident angle. Mathematically, this radiation corresponds to the complex leaky Rayleigh wave pole of the reflection coefficient. The back-reflected amplitude is proportional to the Fourier transform of the incident field on the interface evaluated at the pole wavenumber. The transforms are calculated for several different ultrasonic beam profiles. The beams are characterized by the field on the transmitting transducer. A Gaussian profile is shown to produce a very small back reflection. Similar results hold for a beam composed of an array of Gaussian beams. This beam has the useful property that as limiting cases it reduces to a simple Gaussian beam and a regular SINC beam. Other beam profiles are examined which give a larger effect than these. The role that the radiation function of the transmitter plays in the back reflection is discussed.

Acoustic plane‐wave scattering by a thin rigid plate with a two‐dimensional lattice of rectangular apertures

Scattering of acoustic plane waves by a thin rigid plate with a two‐dimensional lattice of rectangular apertures is solved by a numerical technique that identically satisfies the energy conservation. All fields are expressed as summations of Bloch waves using Floquet's theorem. The normal component of the velocity field in the apertures is expanded in a set of basis functions that have the square root singularity at the edges of the apertures, as dictated by the edge condition. The coefficients of the Bloch waves are expressed in terms of the coefficients of the basis functions, which are in turn solved using Galerkin's method. Reciprocity principle is satisfied to the desired accuracy by taking sufficient number of basis functions and Floquet terms. Wood's anomalies are observed for some Rayleigh critical angles.

Shear modulus gradients in adhesive interfaces as determined by means of ultrasonic Rayleigh waves

Rayleigh surface waves at ultrasonic frequency have been utilized to determine shear modulus variations in the vicinity of adhesive-adherend interfaces. The measurement technique is that of acoustic critical-angle reflectivity conducted at sequentially exposed surfaces of adhesive through the interfacial accommodation zone toward the bondline. Basic theory is presented for determining shear modulus from measured Rayleigh critical angles at adhesive surfaces. Laboratory ultrasonic Rayleigh wave apparatus and its evaluation are described. Experimental procedures for sample preparation and measurement are discussed. Results of Rayleigh critical-angle reflectivity measurements on two different types of adhesives display a decided gradient in shear modulus through the interfacial region. Reference is made to other applications of the ultrasonic Rayleigh wave approach in studying the shear properties near the bondline of adhesive materials.

On the reflection coefficient which exceeds unity

The acoustic reflection coefficient at a lossy liquid–lossless solid interface is shown to be greater than unity near Rayleigh critical angle. This result does not violate the principle of energy conservation. A ray model is presented to explain the underlying physical phenomenon. The peak value of the reflection coefficient is predicted by an approximate expression.

Scattering of an obliquely incident acoustic wave by an infinite cylinder

Expressions are derived for the scattered farfield pressure which results from the illumination of an infinite aluminum cylinder by a plane acoustic wave, whose propagation direction makes an arbitrary angle, α, with the normal to the cylinder axis. Computations are made at angles between α=0 ° and the Rayleigh critical angle, α=31.3 °. Changes in the scattered field are related to shifts in the paths of geometric and elastic surface waves from circumferential at α=0 °, to helical, to axial Rayleigh surface waves generated at the Rayleigh critical angle.

The scattering of an obliquely incident plane acoustic wave from a cylindrical object

A mathematical model is developed to predict the scattering of a plane acoustic wave incident upon an elastic cylinder immersed in water at any angle relative to the cylindrical axis. Solutions to the elastic problem are constructed using scalar and vector potentials. There are three important angular regions of interest. These are at the longitudinal, shear, and Rayleigh critical angles. The scattering amplitude at these angles is derived and will be discussed.

Theory of the effect of stress on ultrasonic plane-wave reflectivity from a water–metal interface

Ultrasonic reflectivity from a solid is a function of longitudinal- and shear-wave velocities and attenuations, as well as the density. Using stress-dependent acoustic velocities obtained from second-order elasticity theory, we have calculated the plane-wave reflectivity at and near the Rayleigh critical angle for uniaxial stress. Reflectivity amplitude and phase results were obtained for both tensile and compressive stress. From the reflectivity amplitude data, values of Raleigh-wave velocity versus stress were obtained. The results exhibit the expected behavior and a comparison with such experimental data as are available shows fair agreement.

Ultrasonic holography at the critical angle

Ultrasonic holography is a recording technique preserving phase and amplitude of a wavefront on a two-dimensional surface. This paper explores the possibility of using this technique to map phase changes occuring on an otherwise featureless surface by illuminating the surface at the Rayleigh critical angle. Since phase response at the critical angle is extremely sensitive to such things as stress concentrations, subsurface defects, or work hardening, it appears that this technique may offer a way to detect and map such distributions. We present preliminary experimental results that substantiate this claim.

Phase Measurements of Reflected Ultrasonic Waves near the Rayleigh Critical Angle

The phase shift with incident angle of reflected ultrasonic waves incident near the Rayleigh critical angle is reported. These measurements substantiate the predictions of our model. These predictions differ substantially from previous theories which neglect attenuation. For frequencies above 15 MHz the slope of the phase shift of an ultrasonic wave reflected from a water-stainless steel boundary reverses at the Rayleigh critical angle. For frequencies below 15 MHz the phase shifts from 0° to +360°; above 15 MHz it shifts from 0° to +60° then to −60° and back to zero again on passing through the Rayleigh critical angle. The sensitivity of this phenomenon suggests that it could be useful in determining or evaluating changes in material properties such as cold work, residual stress, or grain sizes.

Effects of Attenuation on the Rayleigh Critical Angle Reflection Factor

The minimum in the reflection factor for a liquid-to-solid interface at the Rayleigh critical angle has never been adequately explained using classical methods. It will be shown that when attenuation is included in the calculations, by the method of Fitch and Richardson, [Progr. Appl. Mater. Res. 8, 79 (1967)], it is possible to predict this minimum. Experimental results have been obtained that demonstrate very good agreement with this theory. It will also be shown that the principal factor influencing the depth of the minimum is the attenuation of shear waves. Each material possesses a critical frequency for which the reflection factor reaches an absolute minimum, approximately zero. Relationships between this frequency and attenuation parameters are presented. The phase of the reflected wave is also shown to be influenced by attenuation. [This work was supported by the U. S. Atomic Energy Commission under contract.]