Rayleigh waves are well known to attenuate due to scattering when they propagate over a rough surface. Theoretical investigations have derived analytical expressions linking the attenuation coefficient to statistical surface roughness parameters, namely, the surface's root mean squared height and correlation length and the Rayleigh wave's wavenumber. In the literature, three scattering regimes have been identified—the geometric (short wavelength), stochastic (short to medium wavelength), and Rayleigh (long wavelength) regimes. This study uses a high-fidelity two-dimensional finite element (FE) modelling scheme to validate existing predictions and provide a unified approach to studying the problem of Rayleigh wave scattering from rough surfaces as the same model can be used to obtain attenuation values regardless of the scattering regime. In the Rayleigh and stochastic regimes, very good agreement is found between the theory and FE results both in terms of the absolute attenuation values and for asymptotic power relationships. In the geometric regime, power relationships are obtained through a combination of dimensional analysis and FE simulations. The results here also provide useful insight into verifying the three-dimensional theory because the method used for its derivation is analogous.

Elastic waves guided on the surface of a solid are well known to attenuate if the surface is not perfectly flat as a result of scattering.1 The attenuation of Rayleigh waves from rough surfaces has been described analytically, but experimental validations have proved difficult to achieve over a wide range of parameters of the roughness. One reason is that mathematical models are valid in regions that are difficult to replicate in practice because they either require very low roughness or predict high attenuation values, which would render the waves undetectable.

Early work to describe the attenuation of Rayleigh waves resulted in the derivation of expressions for waves on flat surfaces of solids whose material properties induced attenuation. For instance, Maris2 derived expressions for the attenuation of Rayleigh waves on a dielectric crystal with arbitrary crystallographic orientation, where temperature and viscosity were also considered. Their work followed the experimental results of Salzmann et al.,3 who used lasers to measure the effect of the temperature and frequency on the attenuation of Rayleigh waves propagating along quartz crystals. These early studies considered specific attenuation cases related to material properties but did not take into account the roughness that unavoidably exists on all surfaces and causes attenuation even when the waves propagate on perfect lossless elastic materials; the attenuation occurs by partial scattering of the waves from the geometric features of the roughness.1 

Some of the first analytical expressions for the attenuation of Rayleigh waves from rough generalised surfaces were derived by Maradudin and Mills4 and Urazakov and Fal'kovskii.5 The rough surface was described using two statistical parameters—the root mean squared (RMS) height, δ, which describes the amplitude of the peaks and troughs of the roughness, and the correlation length, Λ, which is a measure of the spacing of those peaks. Urazakov and Fal'kovskii used the Rayleigh method for their studies. This method solves the Rayleigh wave equation, which predicts the creation of Rayleigh waves on a stress-free boundary6 but was extended by the authors to apply that stress-free surface condition across the rough surface.

Maradudin and Mills4 used a Green's function approach to solve the relevant equations. Both approaches predict that for the three-dimensional (3D) case, when the roughness is low (for instance, δ / Λ < 0.3), the attenuation coefficient is proportional to the fifth power of the wave's frequency, f, in the region where the Rayleigh wavelength, λR, is much greater than Λ. Maradudin and Mills also demonstrated that the attenuation coefficient is proportional to δ 2. Subsequently, Eguiluz and Maradudin7 published an updated version of their derivations in which the additional scattering of the Rayleigh waves to bulk waves was considered. The new results also demonstrated a proportionality of the attenuation coefficient to f5, as well as a new result of a proportionality to the ratio δ 2 / Λ 2.

Following the work by Eguiluz and Maradudin, de Billy et al.8 completed some experimental work to verify the theory derived in Ref. 7. In their work, attenuation measurements were taken for rough duraluminum and titanium samples, and the effect of varying λR on the attenuation measurements was investigated. The authors found good agreement between their experimental results and the f5 relationship predicted by the theory at the long wavelength limit; however, they observed that this fifth-order proportionality did not hold true at smaller λR values. More recently, Kosachev and Gandurin9 studied the dispersion attenuation of Rayleigh waves on statistically rough hexagonal crystals to expand the work in Ref. 7. In their work, the scattering from the rough surface of an anisotropic hexagonal crystal was theoretically studied with the authors deriving an expression for the attenuation coefficient when the scattering occurs at a generalised crystal orientation. Their expression reduces to the same f5 relationship derived in Ref. 7. for the isotropic case in the long wavelength region. Finally, Chukov10 used the Rayleigh-Born approximation to derive similar relationships to Ref. 7 for the isotropic case, arriving at the same power relationships. In addition, Chukov derived expressions for 2D roughness, which are discussed in more detail below.

However, the theory is restricted to specific δ / Λ and λR combinations. In particular, a large proportion of the theory has been derived in the Rayleigh regime—this is a region where λ R Λ, which is a very low frequency regime. Therefore, the abovementioned considerations motivate two research problems, which this paper attempts to solve—first, to create a finite element (FE) model to validate the existing analytical expressions in the Rayleigh region and, second, to extend the results by FE modelling to other more practically relevant regimes.

To obtain a representative attenuation value for a combination of δ, Λ, and f from numerical modelling, it is necessary to average over a sufficiently large number of attenuation values, which were obtained from individual surfaces characterised by those specific statistical parameters. It is also necessary to have a model sufficiently large to accommodate a representative scattering distance. The implementation of a 3D model would incur a significant computational burden and run-times for such a wide range of parameter values and, therefore, this paper conducts a comprehensive 2D analysis. The validation in two dimensions also infers validation in three dimensions as the same theoretical approach has been implemented for the derivations in both cases.

In addition to the 3D case discussed above, Chukov demonstrated that for the 2D case, the attenuation coefficient is proportional to f4, δ 2, and Λ in the Rayleigh region in comparison with the f 5 δ 2 Λ 2, which has been derived for 3D roughness. Identical power relationships were also derived by Huang and Maradudin,11 using the same small perturbation method implemented in the 3D analysis,7and the f4 relationship was observed experimentally in Ref. 8. These are the proportionality relationships that this paper validates. In addition to this, Maradudin and Eguiluz7 and Huang and Maradudin11 have also derived an expression that gives quantitative values for the attenuation coefficient—the results from the FE simulations in this study are also compared against this expression.

Following this validation, this paper looks into the attenuation coefficient's behaviour in regimes outside of the Rayleigh region. More specifically, both the stochastic ( λ R < Λ) and geometric ( λ R Λ) regimes are investigated. It is worth noting that in the literature, these scattering regimes are studied separately or under different assumptions. In our study, the same approach was used across all three scattering regimes, creating a unified method for studying the scattering of Rayleigh waves from rough surfaces regardless of the scattering regime.

Analytical expressions for the attenuation of Rayleigh waves from statistically rough surfaces are, to our knowledge, very limited in both the geometric and stochastic regions. However, a useful analogous problem has been studied in detail by Van Pamel et al.12 In their study, the authors investigated wave scattering within heterogeneous media—where the wave's propagation was impeded by the presence of grain boundaries within a material. In the Rayleigh regime, the authors found a reduction of the attenuation coefficient dependence by one power of frequency between the 2D and 3D scattering—this is consistent with the reduction of the proportionality from f5 to f4 in two dimensions for roughness scattering, derived by Refs. 10 and 11. The same fourth power proportionality between the attenuation coefficient and frequency has also been derived by Kaganova and Maradudin for the scattering of surface waves in a polycrystalline material.13 Additionally, it was found that at large λR values, belonging to the stochastic region, the attenuation coefficient is proportional to the same powers of δ, Λ, and f regardless of the number of dimensions. Therefore, for the problem studied here, the same power relationships derived for the 3D case by Kosachev et al.14 can be suggested to hold true in two dimensions. Regarding the geometric region, the authors in Refs. 12 and 15 state that the attenuation coefficient is independent of f.

This paper is organized as follows. The theory and analytical results are discussed in more detail in Sec. II. The process of setting up the FE model is described in Sec. III. The results from the FE simulations are presented and discussed in Sec. IV and, finally, Sec. V concludes the work.

This section presents the theory related to both the generation of rough surfaces and power relationships between the attenuation coefficient and different parameters that characterise the incident wave and rough surface. The analytical expressions for calculating the attenuation coefficient, α, along with a brief derivation are also presented and discussed.

The weighted moving average method described in Refs. 1 and 16 was implemented to generate the rough surfaces. An important parameter that characterises a rough surface is its RMS height, δ. This is a measure of the height of the surface's peaks and troughs, relative to a reference surface, whose RMS height is zero. Let x be the direction in which the rough surface lies, and z is the direction perpendicular to x. Now, let h be the distance between the z = 0 line and a point on the rough surface. Using these definitions, the rough surface's height profile can be described by
z = h ( x ) .
(1)
For the rough surfaces used in this study, the mean height of the rough surfaces was set to be zero, i.e., h = 0, where the angled brackets denote the ensemble average value of the quantity. Under this assumption, δ is given by
δ = h 2 .
(2)
A second parameter that is typically used when describing a rough surface is the correlation length, Λ. This can be considered a measure of the spacing between the peaks and troughs of the surface in the x direction. Mathematically, it is defined as the distance over which the correlation function, C(R), drops to 1/e from its initial value for two points separated by a distance R, where the correlation function is defined as
C ( R ) = h ( x ) h ( x + R ) δ 2 .
(3)
In this study, a Gaussian C(R) was chosen as
C ( R ) = exp ( R 2 Λ 2 ) .
(4)
Gaussian roughness was selected for this study as it has been widely studied and is also well understood.1,16,17 Additionally, it has been shown that Gaussian roughness can occur naturally as reported in Refs. 18 and 19, where real fatigue cracks and real surfaces were found to follow a Gaussian roughness profile. Therefore, our choice does not restrict the analysis to an idealised domain.

Initially, a set of random numbers was generated to create the rough surfaces. Then, Eqs. (2) and (4) and the moving average approach described in Ref. 16 were implemented to transform this set of random numbers to a set of correlated numbers, corresponding to the values in the h(x) function.

The usual dispersion relationship for a Rayleigh wave travelling on a flat surface is
ω = C R q ,
(5)
where ω is the Rayleigh wave's angular frequency, CR is the propagation velocity, and q is the wavenumber. Although in most modern work, the wavenumber is denoted by k, the use of q here is in alignment with Maradudin's notation and facilitates the comparison between our results and previous work. When the Rayleigh wave encounters a rough surface, Eq. (5) becomes7 
ω = C R q + Δ ω ,
(6)
where Δ ω is a complex angular frequency perturbation, arising from the presence of the rough surface.
Eguiluz and Maradudin7 have shown that the attenuation length l, which is the length over which the Rayleigh wave's energy falls to 1/e from its initial value, can be calculated via
l 1 = 2 ( δ 2 Λ 2 ) q ω 2 ,
(7)
where ω2 is a function encapsulating the effect of the roughness on the Rayleigh wave. Then, it can be shown that for λ R Λ (Rayleigh region), ω 2 ( ω Λ ) 4, for 3D roughness and, hence, l 3 D R 1 f 5 δ 2 Λ 2, where the subscripts “3D” and “R” denote the presence of 3D roughness and the Rayleigh region, respectively.

An analogous analysis in Refs. 11 and 20 showed that for 2D roughness, the ω2 function is proportional to the third power of ω and Λ in the Rayleigh regime. The derivations were performed using similar methods to those in Ref. 7, which allow ω2 to be directly substituted into the right-hand side of Eq. (7). Therefore, the theory predicts that l 2 D R 1 f 4 δ 2 Λ, where the subscript “2D” denotes the case of two-dimensional (2D) roughness.

For the stochastic region, in an inhomogeneous medium in three dimensions, where λ R < Λ, it has been shown analytically in Ref. 12 that there is no difference in the power relationships in three dimensions and two dimensions between l 1 and the relevant parameters—this observation was subsequently verified numerically by the authors. In this study,12 the authors derived this theory for the attenuation arising from inhomogeneous media—however, their analysis used similar metrics to ours. The size of the “obstacle” causing the attenuation was characterised by its correlation length, and the stochastic regime was defined as the region where q Λ > 1. We can, therefore, suggest the independence of the attenuation length with respect to dimensionality to hold true in our case as well, i.e., the same power relationship exists between l 1 and δ, Λ and f in both 3D and 2D roughness. Kosachev et al.14 have derived an analytical expression for the stochastic region for 3D roughness—therefore, based on their derivation, we expect that l 2 D S 1 f 2 δ 2 Λ 1, where the subscript “S” denotes the stochastic region.

To present power relationships relating to the geometric regime, it is first necessary to introduce the dimensional analysis associated with the asymptotic study of the attenuation coefficient. The asymptotic study is based on the principle of similitude, which stipulates that the study of different phenomena can be treated using equivalent equations if they can be described by the same dimensionless variables.6 For studying attenuation phenomena, an equivalent mathematical analysis can be implemented if roughness parameters (δ and Λ) and the loss in energy (α) are normalised by the Rayleigh wavelength.

More specifically, the asymptotic approximations usually take the form of products of powers for the dimensionless normalised attenuation coefficient, αn,
α n δ n m δ Λ n m Λ ,
(8)
where α n = α λ R , δ n = δ / λ R (normalised RMS height), Λ n = Λ / λ R (normalised correlation length), and m δ and m Λ are the powers of δn and Λn, respectively, to which αn is proportional. Given that unperturbed Rayleigh waves travelling on a smooth flat surface are nondispersive, the frequency is inversely proportional to the wavelength and, therefore Eq. (8) can be rewritten as
α n δ m δ Λ m Λ f m δ + m Λ ,
(9)
or equivalently,
α δ m δ Λ m Λ f m f ,
(10)
where
m f = m δ + m Λ + 1.
(11)

Equation (11) holds true regardless of the number of dimensions and the scattering regime—this can also be confirmed by observing both the 3D and 2D power relationships demonstrated in the previous paragraphs, which all obey Eq. (11). In the geometric regime, the scattering is independent of the frequency.15 Therefore, from Eq. (11), m δ G = 1 m Λ G and l 2 D G 1 δ m δ G Λ 1 m δ G, where the subscript “G” denotes the geometric regime. Here, the power coefficients relating to δ and Λ will be treated as unknowns to be found, and the independence of the attenuation coefficient on f will be validated by the FE model. Finally, it is worth noting that for all of the theoretical derivations, it was assumed that δ < Λ, which follows numerous studies on rough surface scattering in the literature such as Refs. 18 and 21–23.

In our analysis, we will be using a similar attenuation measure to l 1: the attenuation coefficient α, which is defined as the inverse of the distance over which the amplitude of the Rayleigh wave drops to 1/e from its initial value. Using this definition in combination with the fact that the energy of a wave is proportional to the square of its amplitude,24 we can relate α to l by using the following equation:
α = 1 2 l .
(12)

A summary of the expected power relationships between α and δ and between Λ and f is shown in Table I. For ease and uniformity of presentation, we have introduced the dimensionless notation δn, Λn, αn, and β, where α n = α λ R and β = α n Λ / δ. The variable β is defined to later allow us to generate a master curve in which we plot the numerical results against a single variable. If just the conventional normalised attenuation coefficient (αn) is used, this yields results which are functions of both δn and Λn, making it impossible to plot all of the results in a single graph as they are not functions of a single variable. The variable β is defined such that all of the numerical results become a function of a sole variable (δn), assuming that m δ G is zero.

TABLE I.

The expected asymptotic power relationships between the attenuation coefficient and the RMS height, the correlation length and the frequency. Here, q is the wavenumber, δ is the RMS height, Λ is the correlation length, and f is the frequency.

Regime Rayleigh Stochastic Geometric
Limits  q δ < q Λ < 1  q δ < 1 < q Λ  1 < q δ q Λ 
α ( δ , Λ , f )  δ 2 Λ f 4  δ 2 Λ 1 f 2  δ m δ G Λ 1 m δ G 
β ( δ n , Λ n )  δ n Λ n 2  δn  δ n m δ G 1 Λ n m δ G 
Regime Rayleigh Stochastic Geometric
Limits  q δ < q Λ < 1  q δ < 1 < q Λ  1 < q δ q Λ 
α ( δ , Λ , f )  δ 2 Λ f 4  δ 2 Λ 1 f 2  δ m δ G Λ 1 m δ G 
β ( δ n , Λ n )  δ n Λ n 2  δn  δ n m δ G 1 Λ n m δ G 

This section presents the method used to create the FE models, which were used in all subsequent simulations. The purpose of the FE models is to allow us to study the phenomenon of surface wave scattering from a large range of roughness parameters meaningfully and efficiently. Therefore, the necessary steps were taken to minimise the model's size and ensure that the surface wave was of good quality with minimal noise. The process to achieve these properties, as well as the method used to calculate the attenuation coefficient, and the computational resources used are presented below.

Despite the rough surfaces being characterised by their RMS height and correlation length, each surface has a unique h(x) profile. To obtain a meaningful value for the attenuation, it was necessary to perform Monte Carlo simulations and average over a sufficient number of realisations for each δ and Λ to ensure the statistical stability of the result. The simulations were conducted using the high-fidelity, graphics processing unit (GPU)-based FE software package Pogo, which is an explicit time domain FE solver, and visualised using PogoPro.25 Following the discretisation of the domain, Pogo uses the well-known finite difference method with the aid of a stress-free boundary condition to obtain the displacement at each node. If we denote the displacement, velocity, and acceleration matrices of the nodes in the model by U , U ̇ , and U ¨, the equation for elasticity theory becomes
M U ¨ + C U ̇ + KU = F ,
(13)
where M , C , and K are the mass, damping, and stiffness matrices, respectively, and F is the applied force matrix. By implementing the finite difference method and assuming a model with no damping terms, Eq. (13) becomes
M U n + 1 2 U n + U n 1 Δ t 2 + K U n = F ,
(14)
where the superscript “n” denotes the corresponding matrix at the nth time step, and Δ t is the duration of that time step. By rearranging Eq. (14), the displacement at the n + 1 time step can be found by using the values for the displacement at the previous two time steps. The approach described here follows closely the approach taken by other high-fidelity FE studies for elastic wave propagation.26–28 
Inconel 718 was used in all simulations (Young's modulus, E = 208.73 GPa; Poisson's ratio, ν = 0.303; and density, ρ = 7800 kg/ m 3). For a given material, the Rayleigh wavespeed, CR, can be calculated approximately by the formula29 
C R = 0.862 + 1.14 ν 1 + ν C S ,
(15)
where CS is the shear wave speed. For the material parameters defined above, CR was found to be 2892 m/s.
Each rough surface was inserted to form the lower boundary of a 2D rectangular FE domain with the specified material parameters. The length of the rough surface was set to be at least 50Λ to ensure statistical and ergodic stability.1 A Tukey window was applied to the rough surface to ensure a smooth joining with the main material and avoid generating additional artificial attenuation. The Tukey window function, w(x), was of the following form:
w ( x ) = { 0 , | x | L 2 , 1 , | x | L 2 l w 2 , 1 2 [ 1 cos ( 2 π | x | L 2 l w ) ] , otherwise ,
(16)
where L is the length of the rough surface and lw is the length of the window tapering. For our simulations, l w = L / 10 was used. The windowing was implemented by multiplying w(x) with h(x)—this has the effect of smoothing the edges of the surface while leaving the rest of it unaffected. A schematic of the FE model is shown in Fig. 1. 2D triangular elements were used in generating the mesh. The mesh size, Δ x, was set to be approximately equal to λ R / 25, where λR was calculated from the central frequency of each simulation for all models—this is necessary to avoid errors in the elastic wave speed and ensure the numerical stability of the model, which are issues that might arise if the mesh is too coarse, as described in Ref. 30. A similar approach for the mesh size has been used in studies related to ours.31 A typical model size was on the order of 2 × 10 6 degrees of freedom.
FIG. 1.

(Color online) Schematic of the FE model. A Tukey window is applied to the original rough surface, generated using the method described in Sec. II A (in yellow), before it is inserted to form the lower boundary of the FE domain (in red). The scale of the rough surface is exaggerated for better visualisation.

FIG. 1.

(Color online) Schematic of the FE model. A Tukey window is applied to the original rough surface, generated using the method described in Sec. II A (in yellow), before it is inserted to form the lower boundary of the FE domain (in red). The scale of the rough surface is exaggerated for better visualisation.

Close modal

An example of the lower portion of the FE domain after the rough surface was attached to its lower boundary is shown in Fig. 2. There are two interesting features in Fig. 2. First, the ability of Pogo to mesh efficiently can be observed as the irregular mesh only lies close to where the rough surface is located, whereas the mesh efficiently reverts to a regular form in the main bulk of the material as the distance from the rough surface increases. Second, the smooth joining of the rough surface to the FE domain can also be seen. Here, it is worth acknowledging that rough surfaces can be described by fractals32 with past roughness studies using the Weierstrass function,33 which exhibits self-similarity, to model them. Meshing unavoidably truncates this fractal feature. However, it is expected that the absence of this fractal nature will not affect the result as geometrical features significantly smaller than the wavelength cannot be resolved by the wave.22,34

FIG. 2.

(Color online) The detail of the FE domain's meshing after the Tukey-windowed rough surface has been applied to its lower boundary.

FIG. 2.

(Color online) The detail of the FE domain's meshing after the Tukey-windowed rough surface has been applied to its lower boundary.

Close modal

In the FE model, the input signal used was a five-cycle Hann windowed tone burst. The absolute value of signal's amplitude was arbitrarily selected because the simulation is linear and we are concerned about ratios of measured results and not their absolute values. A source line, comprised of multiple source nodes, was located to the left of the rough surface. For each simulation, the size of the source was set to be equal to three Rayleigh wavelengths, calculated from the simulation's central frequency. To obtain a Rayleigh wave travelling toward the rough surface, a phase delay was applied to each node such that constructive interference from the signal from each node occurred in the desired direction. This method was implemented as follows:

  • Two sinusoidal time domain signals were created with a 90° phase shift between them.

  • Pogo provides the ability to assign each source node a unique amplitude, which scales the time domain signal assigned to that node accordingly. Therefore, the amplitude at each node was selected in a way such that a clean Rayleigh wave with the correct amplitude and phase was created by the interference of the signal from all of the nodes.

  • The complex amplitude assigned the ith source node, ai, located at the xi position in the model is given by
    a i = 1 2 [ 1 cos ( 2 π x i x min x max x min ) ] e j q x i ,
    (17)

    where x min is the position of the leftmost source node, x max is the position of the rightmost source node, and j is the imaginary unit. The collective use of a unique amplitude at each node, according to Eq. (17), results in the constructive interference of the signals from all of the source nodes in the correct direction, which generates a clean Rayleigh wave.

  • The first signal was applied to the ith node with a weighting of [Im(ai),-Re(ai)] and the second signal was applied with a weighting of [-Re(ai),-Im(ai)], where the two entries in the previous vectors denote the x and z direction, respectively, and the notations Re( ) and Im( ) denote the real and imaginary parts of their argument, respectively. The amplitudes of the x and z components of the Rayleigh wave are arbitrary because one is interested in the ratio of the amplitudes before and after the rough surface rather than the absolute values.

An example of a Rayleigh wave created using the method described above is shown in Fig. 3. As shown in Fig. 3, a pure Rayleigh wave is created by implementing this method. The minimal secondary waves that exist in Fig. 3 just above the right-hand end of the Rayleigh wave lie away from the rough surface and, therefore, do not interfere with the attenuation measurements.

FIG. 3.

(Color online) An example of a Rayleigh wave field, travelling in the positive x direction created using the method described in Sec. III. The color scale represents the absolute magnitude of the displacement at each node. The Rayleigh wave's centre frequency is 6 MHz and the rough surface has δ = 25 μm and Λ = 50 μm.

FIG. 3.

(Color online) An example of a Rayleigh wave field, travelling in the positive x direction created using the method described in Sec. III. The color scale represents the absolute magnitude of the displacement at each node. The Rayleigh wave's centre frequency is 6 MHz and the rough surface has δ = 25 μm and Λ = 50 μm.

Close modal
To obtain the information required to calculate the attenuation coefficient, two monitor nodes were used in the model, one on either side of the rough surface. The Rayleigh wave's z-amplitude was measured at each monitor node. The attenuation coefficient was then calculated by our definition of α,
α = 1 x d ln ( A 2 A 1 ) ,
(18)
where A1 and A2 are the amplitudes of the Fourier transforms of the Rayleigh wave before and after the rough surface, and xd is the distance between the locations at which A1 and A2 were obtained.

Finally, to avoid unwanted noise from the source or scattered waves interfering with the attenuation measurements, absorbing layer regions were applied to the top, left, and right sides of the FE domain. These regions are defined by material parameters whose damping gradually increases, leading to attenuation rather than reflection of the waves.35 The addition of absorbing layers in similar setups has proven to be beneficial in related rough surface studies.36 

For each Monte Carlo simulation, 100 unique surfaces and, hence FE domains, were generated. Using 1 Nvidia GTX 1080Ti (Santa Clara, CA) with 11 GB of memory, each set of 100 models takes about 1.5 h to complete. This efficiency allowed one to run several statistically stable sets of simulations, covering a wide range of roughness, which better validates the theoretical models. A 3D approach would not have been able to achieve such wide variety of simulation parameter values due to computational limitations. The 2D study presented here provides useful and meaningful insight in the verification of the existing 3D theory as the 2D11 and 3D7 theories have been derived under the same assumptions using similar methods. Additionally, the 2D work here identifies the most important regimes in which 3D investigations may be conducted. It is also worth noting that the same method described here was used to generate all of the FE models required for our study, regardless of the scattering regime. This strengthens the universality of our findings as it has eliminated the need to study each regime separately, which is often the case in the literature.

This section presents the results from our FE simulations and is split into three parts. In the first part, our FE results are compared quantitatively with Eq. (7) to investigate the agreement between the theory and FE model. After this agreement is established, the second part presents the results relating to the power relationships presented in Table I.

The initial roughness statistical parameters used for the Rayleigh regime were selected to reflect values of roughness that can be found on metal parts created by additive manufacturing. More specifically, values of δ in the range of 10–25 μm37–39 were chosen. The correlation length values were then adopted such that they fulfilled the δ < Λ condition as required by the limiting conditions in Table I. When the investigation relating to the Rayleigh regime was completed, we explored the stochastic and geometric regimes by expanding our initial selection of δ and Λ values.

Each surface realisation, despite being characterised by its δ and Λ values, has a unique profile due to the inherent randomness of roughness. It is, therefore, necessary to average over a sufficient number of realisations to get results that are statistically meaningful. Each datapoint in Figs. 5–9 is an ensemble average α value, obtained from the Monte Carlo simulation of 100 realisations. This number of realisations lies within the range of 50–200, which has been used in similar studies22,26,31 but has also been verified for its statistical stability for our specific study by conducting a convergence analysis similar to that of Ref. 40. An example of the convergence plots for three roughness cases is shown in Fig. 4.

FIG. 4.

(Color online) The variation of the attenuation coefficient as the number of realisations increases for three roughness scenarios at f = 10 MHz.

FIG. 4.

(Color online) The variation of the attenuation coefficient as the number of realisations increases for three roughness scenarios at f = 10 MHz.

Close modal
FIG. 5.

(Color online) The comparison of theoretical α values with FE results. The attenuation coefficient is plotted against the dimensionless quantity q Λ, which is analogous to plotting against the frequency. The theoretical predictions of the model of Ref. 11 are shown using the curve, and the FE results are plotted as “×.”

FIG. 5.

(Color online) The comparison of theoretical α values with FE results. The attenuation coefficient is plotted against the dimensionless quantity q Λ, which is analogous to plotting against the frequency. The theoretical predictions of the model of Ref. 11 are shown using the curve, and the FE results are plotted as “×.”

Close modal
FIG. 6.

(Color online) The FE results relating to the Rayleigh regime. The FE results, plotted as “×,” and the line of best fit through them are shown. The gradient of the best fit line, m, is also shown. Values of the attenuation coefficient (either absolute or normalised) are plotted on the vertical axis, whereas the variable whose power relationship is investigated is plotted on the horizontal axis.

FIG. 6.

(Color online) The FE results relating to the Rayleigh regime. The FE results, plotted as “×,” and the line of best fit through them are shown. The gradient of the best fit line, m, is also shown. Values of the attenuation coefficient (either absolute or normalised) are plotted on the vertical axis, whereas the variable whose power relationship is investigated is plotted on the horizontal axis.

Close modal
FIG. 7.

(Color online) The FE results relating to the stochastic regime. The FE results, plotted as “×,” and the line of best fit through them are shown. The gradient of the best fit line, m, is also shown. Values of the attenuation coefficient (either absolute or normalised) are plotted on the vertical axis, whereas the variable whose power relationship is investigated is plotted on the horizontal axis.

FIG. 7.

(Color online) The FE results relating to the stochastic regime. The FE results, plotted as “×,” and the line of best fit through them are shown. The gradient of the best fit line, m, is also shown. Values of the attenuation coefficient (either absolute or normalised) are plotted on the vertical axis, whereas the variable whose power relationship is investigated is plotted on the horizontal axis.

Close modal
FIG. 8.

(Color online) αn vs Λ plot for the geometric regime. The FE results, plotted as “×,” and the line of best fit through them are shown. The gradient of the best fit line, m, is also shown. Values of the attenuation coefficient (either absolute or normalised) are plotted on the vertical axis, whereas the variable whose power relationship is investigated is plotted on the horizontal axis.

FIG. 8.

(Color online) αn vs Λ plot for the geometric regime. The FE results, plotted as “×,” and the line of best fit through them are shown. The gradient of the best fit line, m, is also shown. Values of the attenuation coefficient (either absolute or normalised) are plotted on the vertical axis, whereas the variable whose power relationship is investigated is plotted on the horizontal axis.

Close modal
FIG. 9.

(Color online) β vs δn master plot. Here, generalised attenuation coefficient values are drawn for a large combination of frequencies and roughness parameters. The datapoints cover both the stochastic and geometric regions, where the above proposed ad hoc approximation predicts that β δ n and β δ n 1, respectively, with the transition occurring at the δ n = 1 / 2 π point. The Monte Carlo results with a fixed Λ ( = 800 μ m ) are plotted as “×,” the results with a common f (=1 MHz) are plotted as “ ,” and the results with a fixed δ ( = 200 μ m ) are plotted as “⋄.” The dashed black lines show the asymptotic approximation for both scattering regimes, whereas the red dashed line indicates the transition point between them.

FIG. 9.

(Color online) β vs δn master plot. Here, generalised attenuation coefficient values are drawn for a large combination of frequencies and roughness parameters. The datapoints cover both the stochastic and geometric regions, where the above proposed ad hoc approximation predicts that β δ n and β δ n 1, respectively, with the transition occurring at the δ n = 1 / 2 π point. The Monte Carlo results with a fixed Λ ( = 800 μ m ) are plotted as “×,” the results with a common f (=1 MHz) are plotted as “ ,” and the results with a fixed δ ( = 200 μ m ) are plotted as “⋄.” The dashed black lines show the asymptotic approximation for both scattering regimes, whereas the red dashed line indicates the transition point between them.

Close modal

As shown in Fig. 4, the number of realisations required to obtain a converged α value is a function of the surface roughness. For the case with the longest correlation length in Fig. 4, the α value converges after approximately 15 realisations, whereas for the other 2 cases, where the correlation length is shorter and, therefore, the peaks and troughs of the surface are closer together, a higher number of realisations is required for convergence. However, all cases converged before the 100 realisation limit was reached, further supporting our choice.

Finally, following the verification of the quantitative and asymptotic results, the last part of this section presents a “master” attenuation curve on which attenuation values from various δ, Λ, and f combinations are plotted as a method to further verify the agreement of the FE and theory over a wider range of parameters. A summary of all of the power relationships obtained by FE modelling is also given.

A comparison between the theoretical α and values predicted by combining Eqs. (7) and (12) is shown in Fig. 5. The statistical parameters used to obtain Fig. 5 were δ = 200 μ m, Λ = 800 μ m, and the frequency was varied from 0.4 to 1.75 MHz.

In the horizontal axis of Fig. 5, the dimensionless quantity q Λ is plotted—this is analogous to plotting against the frequency as in all of the simulations, Λ was fixed, and the wavenumber q varies linearly with the frequency. Presenting the results as such also follows the work of Huang and Maradudin,11 who also plot against q Λ. The mean attenuation coefficient value is plotted as “×,” and the plot includes error bars whose lengths are equal to ±2 standard errors (SEs), where SE is defined as
SE = σ n .
(19)
In Eq. (19), σ is the standard deviation and n is the number of realisations.

It is clear that the FE results follow the curve predicted by the theory. The agreement is also evident in more than one of the scattering regimes described in Sec. II B because Fig. 5 contains results for q Λ values both greater and smaller than one, which is defined as the value at which the scattering behaviour changes from the Rayleigh to the stochastic regime. Using the definitions in Table I, the results of Fig. 5 cover the Rayleigh and stochastic regions with some of the points at higher q Λ values lying in the transition region between the stochastic and geometric regimes. It is worth noting that there are some very small discrepancies between the theoretical attenuation coefficient and FE results. However, we are looking at differences between two approaches here, so it would be inappropriate to view this as errors in the FE simulation not matching the theory. We believe there are three possible sources of discrepancies—approximations in the theory, errors in the FE simulations, and insufficient convergence of the attenuation coefficient at 100 realisations. The outcome of the recent studies using high-fidelity FE analysis is that the results from the FE modelling are highly accurate. The authors in Refs. 26–28 discuss in detail the high degree of accuracy achieved through FE, hence, the error associated with the FE approach here is expected to be very small. Regarding the insufficient convergence issue, the results in Fig. 4 show good convergence, hence, we expect this error to also be small. Therefore, it is possible, indeed likely, that the approximations in the theory are a bigger contributor to the differences between the theory and FE results. This confirms the validity of our FE model and allows us to proceed with a deeper analysis of each scattering regime.

In this section, we present the results relating to the power relationships in Table I. To calculate the power relationship between α and the variable of interest, the sought power coefficient was determined numerically by a least squares regression analysis, and the results are plotted on a log-log scale in which the power coefficient of the best fitting power function is represented by the slope of the regression line. Similar to Fig. 5, SE bars have been added to all of the plots relating to the power relationships.

1. Rayleigh regime

The results relating to the Rayleigh region are shown in Fig. 6. We are plotting against the characteristic parameters f, δn, and Λn, which are the frequency and normalised RMS height and correlation length, respectively, as defined in Sec. II B. According to the theory presented in Sec. II B, we are expecting α f 4 δ 2 Λ in the Rayleigh regime. Figure 6(a) shows the simulation data relating to the f4 relationship. To generate the datapoints in Fig. 6(a), the rough surfaces were defined to have δ = 10 μm and Λ = 20 μ m. The frequency was varied from 1.75 to 2.25 MHz, and at each frequency point, a Monte Carlo simulation of 100 realisations was completed. The gradient of the best fit line in Fig. 6(a) is 3.77, which is close to the expected value of four.

Figure 6(b) shows the simulation results relating to the δ 2 relationship. To produce Fig. 6(b), the frequency of the simulations was set to 0.5 MHz ( λ R = 5800 μ m), and Λ was set to 80 μm. Then, δ was varied from 30 to 80 μm. The gradient of the best fit line in Fig. 6(b) is 1.77, which is fairly close to two. It is worth noting that the power relationship here is calculated for α n ( δ n ), however, it holds true against δ as well—this is because the simulations were completed at a fixed frequency (hence, wavelength) and, therefore, the values in Fig. 6(b) have been normalised by the same scalar.

Figure 6(c) shows the simulation results relating to the Λ relationship. To produce Fig. 6(c), the frequency of the simulations was set to 0.5 MHz ( λ R = 5800 μ m) and δ was set to 20 μm. Then, Λ was varied from 50 to 90 μm. The gradient of the best fit line in Fig. 6(c) is 0.94. Again, although the power relationship has been calculated for α n ( Λ n ), it remains the same for Λ for the same reason explained in the previous paragraph.

It appears that our FE results match the expected power relationships presented in Table I very closely. The power relationships can be explained as follows. The analysis in Maradudin and Huang11 demonstrated that the ω2 function is proportional to ( ω Λ ) 3 as q Λ tends to zero. When this is substituted into Eq. (7), it yields the α f 4 δ 2 Λ relationship, demonstrated by our FE results here. Physically, the attenuation coefficient tends to zero at low frequencies because the Rayleigh wavelength becomes so long relative to the statistical parameters characterising the surface that the surface appears flat to the wave.

2. Stochastic regime

The results relating to the stochastic region are shown in Fig. 7. Based on the theory presented in Sec. II B, a proportionality of α to f 2 δ 2 Λ 1 is expected in the stochastic regime.

Figure 7(a) shows the FE results relating to the f2 relationship. For this set of simulations, δ was set to 200 μm and Λ was set to 800 μm. The gradient of the best fit line is 1.94. To produce Fig. 7(b), the frequency of all simulations was set to 1 MHz ( λ R = 2900 μ m) and Λ was set to 800 μm. Then, δ was varied from 100 to 400 μm, satisfying the stochastic region's condition. The gradient of the best fit line was found to be 2.15. Finally, Fig. 7(c) was generated by setting the frequency again to 1 MHz. Then, δ was fixed to 200 μm and Λ was varied from 400 to 1600 μm. The gradient of the best fit line was found to be −1.25.

Overall, it appears that our FE model follows the already established theory in the stochastic region well. The power relationships here can be again explained by the behaviour of the ω2 function in the stochastic region. As has been demonstrated in Ref. 11, in the stochastic region, ω 2 ω Λ. When this relationship is substituted in Eq. (7), it yields the f 2 δ 2 Λ 1 proportionality predicted by the theory and supported by our FE results.

3. Geometric regime

The results relating to the geometric region are shown in Fig. 8. The geometric region is a region in which the RMS height is greater than λ R / 2 π as per the definition made in Table I. Therefore, the frequency in this set of simulations was set to 5 MHz ( λ R = 580 μ m). Then, δ was set to 200 μm and Λ was varied from 800 to 1600 μm. The gradient of the best fit line in Fig. 8 is −0.83.

The attenuation coefficient can be seen to be decreasing with an increase in the frequency in Fig. 8. Physically, in this region, the wavelength has become so small compared with the correlation length that the roughness does not impede its motion—the wave travels along the peaks and troughs without being scattered. Additionally, from the dimensional analysis presented in Sec. II B, the fact that m Λ G is approximately equal to one implies that m δ G is approximately equal to zero in the geometric regime. This is further investigated and validated in Sec. IV C.

In Table I, we have also introduced the generalised attenuation coefficient, β. Using this allows us to further verify the validity of the theory in the stochastic and geometric regions, by plotting a wider range of Monte Carlo results against only the variable δn. As shown in Fig. 9, the results follow the asymptotic approximation lines independently of which parameter was the variable in each FE Monte Carlo set. Additionally, the transition between the stochastic and geometric region can clearly be seen at δ n = 1 / 2 π. This is the expected location of the transition and can be derived by identifying that in Table I, the transition point between the stochastic and geometric regimes is defined to be where q δ = 1.

It is now worth noting that the gradient of the asymptote on the right-hand side of the plot, which corresponds to the geometric regime, is equal to −1. Therefore, from our dimensional analysis and the values in the last row of Table I, m δ G is again found to be equal to zero because m δ G − 1 = −1. This also confirms that m Λ G = 1, which is demonstrated by our FE results, indicates that the attenuation coefficient is independent of the frequency in the geometric regime, because m f G must be approximately equal to zero from Eq. (11). A summary of the power relationships obtained from the FE results in all scattering regimes is shown in Table II.

TABLE II.

The expected asymptotic power relationships between the attenuation coefficient and the RMS height, the correlation length and the frequency. Here, q is the wavenumber, δ is the RMS height, Λ is the correlation length, and f is the frequency.

Regime Rayleigh Stochastic Geometric
Limits  q δ < q Λ < 1  q δ < 1 < q Λ  1 < q δ q Λ 
α ( δ , Λ , f )  δ 1.77 Λ 0.94 f 3.77  δ 2.15 Λ 1.25 f 1.94  Λ 0.83 
Regime Rayleigh Stochastic Geometric
Limits  q δ < q Λ < 1  q δ < 1 < q Λ  1 < q δ q Λ 
α ( δ , Λ , f )  δ 1.77 Λ 0.94 f 3.77  δ 2.15 Λ 1.25 f 1.94  Λ 0.83 

Comparing Tables I and II, it is clear that there is good agreement in the asymptotic power relationship coefficient across all scattering regimes. This has two implications. First, we have managed to verify the well-established theory regarding scattering in the Rayleigh regime both quantitatively and asymptotically. Second, our FE model was able to also verify the asymptotic in the stochastic and geometric regimes, confirming the applicability of assumptions from scattering12 to our study.

A comprehensive study of the attenuation of Rayleigh waves from 2D statistically rough surfaces using FE modelling has been presented. Three distinct scattering regimes have been identified from the literature—the Rayleigh (low frequency), stochastic (low to medium frequency), and geometric (high frequency) regimes. Analytical formulas, predicting attenuation values, have been derived in the past,11 as well as asymptotic power relationships10,11 between α and δ and between Λ and f.

Here, we attempted to validate the existing theory using the FE analysis and extend the results to regions in which the theory is less established or obtain results with a wider combination of δ, Λ, and f values. We have found good agreement between the theory and FE results in all three regimes—in the Rayleigh and stochastic regimes, good agreement was found both quantitatively and asymptotically, and the f4 relationship between α and the frequency in the Rayleigh regime was also observed. For the geometric regime, power relationships were derived by a combination of FE modelling and dimensional analysis.

The FE model's ability to follow the theory creates a plethora of useful implications. The theoretical formulas rely heavily on the ω2 function, which is a complicated function comprising multiple subfunctions, many of which have a different form, depending on the region of interest, meaning that calculating ω2 is far from straightforward. The ω2 function's validity is also limited in terms of the roughness parameters for which it can produce results ( δ / Λ < 0.3), and its behaviour has not been studied extensively in the literature in the geometric regime. FE modelling removes the necessity to obtain this function and allows for direct calculation of the attenuation coefficient. Additionally, the FE models can potentially be extended to regimes where the literature is more limited, such as the geometric regime, and can also simulate δ and Λ parameters outside the theory's region of validity. Finally, the use of FE modelling has provided a more unified approach to the study of rough surface scattering. In the literature, each scattering regime is largely studied in depth on its own, whereas the FE approach here has been able to verify the theory in all three regimes by always implementing the same method.

Finally, it is worth noting that despite the results here being obtained from 2D simulations, they are still relevant for the 3D analytical formulas. The mathematical approach used to derive the 3D7 and 2D11 theories are analogous—therefore, the FE validation of the 2D theory in our study provides important insight for the 3D theory, indicating that it will also hold true for FE simulations and experimental scenarios.

G.S. is funded by the United Kingdom (UK) Research Centre in nondestructive evaluation (NDE), iCASE No. 17000191, with contributions from Rolls-Royce Holdings plc and Jacobs Engineering Group Inc. During this work. P.H. was partially funded by the UK Engineering and Physical Sciences Research Council (EPSRC) Fellowship No. EP/M020207/1. M.L. is partially sponsored by the EPSRC.

1.
J. A.
Ogilvy
,
Theory of Wave Scattering from Random Rough Surfaces
(
CRC Press
,
New York
,
1991
), pp.
1
37
.
2.
H. Z.
Maris
, “
Attenuation of ultrasonic surface waves by phonon viscosity and heat conduction
,”
Phys. Rev.
188
(
3
),
1308
1311
(
1969
).
3.
E.
Salzmann
,
T.
Plieninger
, and
K.
Dransfeld
, “
Attenuation of elastic surface waves in quartz at frequencies of 316 MHz and 1047 MHz
,”
Appl. Phys. Lett.
13
(
14
),
14
1311
(
1968
).
4.
A. A.
Maradudin
and
D. L.
Mills
, “
The attenuation of Rayleigh surface waves by surface roughness
,”
Ann. Phys
100
(
1-2
),
262
309
(
1976
).
5.
E. I.
Urazakov
and
L. A.
Fal'kovskii
, “
Propagation of a Rayleigh wave along a rough surface
,”
Sov. Phys. - JETP
36
(
6
),
1214
1216
(
1972
).
6.
J. W. S.
Rayleigh
,
The Theory of Sound
(
Macmillan
,
London
,
1877
), pp.
1
370
.
7.
A. G.
Eguiluz
and
A. A.
Maradudin
, “
Frequency shift and attenuation length of a Rayleigh wave due to surface roughness
,”
Phys. Rev. B
28
(
2
),
728
747
(
1983
).
8.
M.
de Billy
,
G.
Quentin
, and
E.
Baron
, “
Attenuation measurements of an ultrasonic Rayleigh wave propagating along rough surfaces
,”
J. Appl. Phys.
61
,
2140
2145
(
1987
).
9.
V. V.
Kosachev
and
Y. N.
Gandurin
, “
Rayleigh wave dispersion and attenuation on a statistically rough free surface of a hexagonal crystal
,”
Phys. Solid State
45
(
2
),
391
399
(
2003
).
10.
V.
Chukov
, “
Rayleigh wave scattering by statistical arbitrary form roughness
,”
Solid State Commun.
149
(
47-48
),
2219
2224
(
2009
).
11.
X.
Huang
and
A. A.
Maradudin
, “
Propagation of surface acoustic waves across random gratings
,”
Phys. Rev. B
36
(
15
),
7827
7839
(
1987
).
12.
A.
Van Pamel
,
P. B.
Nagy
, and
M. J. S.
Lowe
, “
On the dimensionality of elastic wave scattering within heterogeneous media
,”
J. Acoust. Soc. Am.
140
(
6
),
4360
4366
(
2016
).
13.
I. M.
Kaganova
and
A. A.
Maradudin
, “
Surface acoustic waves on a polycrystalline substrate
,”
Phys. Scr.
T44
,
104
112
(
1992
).
14.
V. V.
Kosachev
,
Y. V.
Lokhov
, and
V. N.
Chukov
, “
Theory of attenuation of Rayleigh surface acoustic waves on a free randomly rough surface of a solid
,”
Sov. Phys. - JETP
67
(
9
),
1825
1830
(
1988
).
15.
F. E.
Stanke
and
G. S.
Kino
, “
A unified theory for elastic wave propagation in polycrystalline materials
,”
J. Acoust. Soc. Am.
75
(
3
),
665
681
(
1984
).
16.
J. A.
Ogilvy
, “
Computer simulation of acoustic wave scattering from rough surfaces
,”
J. Phys. D: Appl. Phys.
21
(
2
),
260
267
(
1988
).
17.
J. A.
Ogilvy
and
J. R.
Foster
, “
Rough surfaces: Gaussian or exponential statistics?
,”
J. Phys. D: Appl. Phys
22
(
9
),
1243
1251
(
1989
).
18.
J.
Zhang
,
B. W.
Drinkwater
, and
P. D.
Wilcox
, “
Longitudinal wave scattering from rough crack-like defects
,”
IEEE Trans. Ultrason., Ferroelectr., Freq. Control
58
(
10
),
2171
2180
(
2011
).
19.
W.
Choi
,
F.
Shi
,
M. J. S.
Lowe
,
E. A.
Skelton
,
R. V.
Craster
, and
W. L.
Daniels
, “
Rough surface reconstruction of real surfaces for numerical simulations of ultrasonic wave scattering
,”
NDT E Int.
98
,
27
36
(
2018
).
20.
V. V.
Kosachev
and
A. V.
Shchergov
, “
Dispersion and attenuation of surface acoustic waves of various polarisations on a stress-free randomly rough surface of solid
,”
Ann. Phys.
240
(
2
),
225
265
(
1995
).
21.
S. G.
Haslinger
,
M. J. S.
Lowe
,
P.
Huthwaite
,
R. V.
Craster
, and
F.
Shi
, “
Elastic shear wave scattering by randomly rough surfaces
,”
J. Mech. Phys. Solids
137
,
103852
(
2019
).
22.
J.
Zhang
,
B. W.
Drinkwater
, and
P. D.
Wilcox
, “
Effect of roughness on imaging and sizing rough crack-like defects using ultrasonic arrays
,”
IEEE Trans. Ultrason., Ferroelectr., Freq. Control
59
(
5
),
939
948
(
2012
).
23.
F.
Shi
,
M. J. S.
Lowe
, and
R. V.
Craster
, “
Diffusely scattered and transmitted elastic waves by random rough solid-solid interfaces using an elastodynamic Kirchhoff approximation
,”
Phys. Rev. B
95
(
21
),
214305
(
2017
).
24.
J. L.
Rose
,
Ultrasonic Guided Waves in Solid Media
(
Cambridge University Press
,
New York
,
2014
).
25.
P.
Huthwaite
, “
Accelerated finite element elastodynamic simulations using the GPU
,”
J. Comput. Phys.
257
(
Part A
),
687
707
(
2014
).
26.
S. G.
Haslinger
,
M. J. S.
Lowe
,
P.
Huthwaite
,
R. V.
Craster
, and
F.
Shi
, “
Appraising Kirchhoff approximation theory for the scattering of elastic shear waves by randomly rough defects
,”
J. Sound Vib.
460
,
114872
(
2019
).
27.
M.
Huang
,
G.
Sha
,
P.
Huthwaite
,
S. I.
Rokhlin
, and
M. J. S.
Lowe
, “
Maximizing the accuracy of finite element simulation of elastic wave propagation in polycrystals
,”
J. Acoust. Soc. Am.
148
(
4
),
1890
1910
(
2020
).
28.
A. A. E.
Zimmermann
,
P.
Huthwaite
, and
B.
Pavlakovic
, “
High-resolution thickness maps of corrosion using SH1 guided wave tomography
,”
Proc. R. Soc. A
477
,
20200380
(
2021
).
29.
J. D.
Achenbach
,
Wave Propagation in Elastic Solids
(
North Holland
,
Amsterdam
,
1973
), p.
192
.
30.
M. B.
Drozdz
, “
Efficient finite element modelling of ultrasound waves in elastic media,” Ph.D. dissertation
,
Imperial College of Science Technology and Medicine
,
London
,
2008
.
31.
F.
Shi
,
W.
Choi
,
M. J. S.
Lowe
,
E. A.
Skelton
, and
R. V.
Craster
, “
The validity of Kirchhoff theory for scattering of elastic waves from rough surfaces
,”
Proc. R. Soc. A
471
,
20140977
(
2015
).
32.
B. B.
Mandelbrot
,
The Fractal Geometry of Nature
(
Freeman and Co. Ltd
.,
San Francisco
,
1982
), pp.
1
460
.
33.
D. L.
Jaggard
and
Y.
Kim
, “
Diffraction by band-limited fractal screens
,”
J. Opt. Soc. Am.
4
(
6
),
1055
1062
(
1987
).
34.
J. B.
Elliott
,
P.
Huthwaite
,
M. J. S.
Lowe
,
R.
Phillips
, and
D. J.
Duxbury
, “
Sizing subwavelength defects with ultrasonic imagery: An assessment of super-resolution imaging on simulated rough defects
,”
IEEE Trans. Ultrason., Ferroelectr., Freq. Control
66
(
10
),
1634
1648
(
2019
).
35.
P.
Rajagopal
,
M. B.
Drozdz
,
E. A.
Skelton
,
M. J. S.
Lowe
, and
R. V.
Craster
, “
On the use of absorbing layers to simulate the propagation of elastic waves in unbounded isotropic media using commercially available finite element packages
,”
NDT E Int.
51
,
30
40
(
2012
).
36.
W.
Hassan
,
M.
Blodgett
, and
S.
Bondok
, “
Numerical analysis of the Rayleigh wave dispersion due to surface roughness
,”
AIP Conf. Proc.
700
,
262
269
(
2004
).
37.
A.
Boschetto
,
L.
Bottini
, and
F.
Veniali
, “
Surface roughness and radiusing of Ti6Al4V selective laser melting-manufactured parts conditioned by barrel finishing
,”
Int. J. Adv. Manuf. Technol.
94
,
2773
2790
(
2018
).
38.
V.
Alfieri
,
P.
Argenio
,
F.
Caiazzo
, and
V.
Segi
, “
Reduction of surface roughness by means of laser processing over additive manufacturing metal parts
,”
Materials
10
(
1
),
30
42
(
2016
).
39.
A.
Boschetto
,
L.
Bottini
, and
F.
Veniali
, “
Roughness modeling of AlSi10Mg parts fabricated by selective laser melting
,”
J. Mater. Process. Technol.
241
,
154
163
(
2017
).
40.
J. R.
Pettit
,
A. E.
Walker
, and
M. J. S.
Lowe
, “
Improved detection of rough defects for ultrasonic nondestructive evaluation inspections based on finite element modeling of elastic wave scattering
,”
IEEE Trans. Ultrason., Ferroelectr., Freq. Control
62
(
10
),
1797
1808
(
2015
).
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