# Acoustic Scattering Models from Rough Surfaces: A Brief Review and Recent Advances

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

^{2}/L (σ and L being the RMS surface height and the height correlation length). Kirchhoff is considered accurate when this parameter is smaller than an upper bound c depending on the wavelength: (σ

^{2}/L < c). Kirchhoff accuracy is also dependent on incidence and scattering angles. In the recent years new developments or comparisons of the previous methods have also been proposed in the literature and for different applications, such as non-destructive evaluation (NDE), underwater acoustics or even medicine.

## 2. Rough Surfaces Parametrization

#### 2.1. Statistical Properties

**r**+ $R$ away. Gaussian functions are often used as an approximation of the correlation of real surfaces. Assuming the same Gaussian correlation in every direction, C can be written:

#### 2.2. Generation Process of an Individual Gaussian Rough Surface

#### 2.3. Implementation Example for Generation of an Individual Gaussian Rough Surface

- -
- The RMS surface height: $\sigma ={\left[\langle {h}^{2}\rangle \right]}^{\frac{1}{2}}$. As the height is defined with zero mean ($\langle h\rangle $ = 0), the standard deviation $\sigma ={\left[\langle {\left(h-\langle {h}^{2}\rangle \right)}^{2}\rangle \right]}^{\frac{1}{2}}$ of the height is equal to the RMS height $rms(h)={\left[\langle {h}^{2}\rangle \right]}^{\frac{1}{2}}$: $\sigma ={\left[\langle {\left(h-\langle h\rangle \right)}^{2}\rangle \right]}^{\frac{1}{2}}={\left[\langle {h}^{2}\rangle \right]}^{\frac{1}{2}}=rms(h)$.
- -
- The correlation lengths, ${L}_{x}^{}$ and ${L}_{y}^{}$, along the local axes x and y.

## 3. Historical Approximations

#### 3.1. The Perturbation Theory

#### 3.2. The Kirchhoff Approximation

#### 3.2.1. Theory and History

- 1.
- Use of the Helmholtz equation: the scattered field is:$$\psi \left(r\right)={\psi}^{inc}\left(r\right)+{{\displaystyle \int}}_{{S}_{0}}\left(\psi \left({r}_{0}\right)\frac{\partial G\left(r,{r}_{0}\right)}{\partial {n}_{0}}-G\left(r,{r}_{0}\right)\frac{\partial \psi \left({r}_{0}\right)}{\partial {n}_{0}}\right)d{S}_{0}.$$

- 2.
- Boundary Conditions on the rough surface: Ogilvy [27] considered for instance Dirichlet boundary conditions (soft case) so that $\psi ({r}_{0})$ = 0. So one term in the previous Green integral disappears:$$\psi \left(r\right)={\psi}^{inc}\left(r\right)+{{\displaystyle \int}}_{{S}_{0}}\left(-G\left(r,{r}_{0}\right)\frac{\partial \psi \left({r}_{0}\right)}{\partial {n}_{0}}\right)d{S}_{0}$$
- 3.
- Kirchhoff approximation: the tangential plane approximation enables to approximate the value of $\frac{\partial \psi \left({r}_{0}\right)}{\partial {n}_{0}}$ inside the previous integral.
- 4.
- Far field approximation: the integrand value can be simplified by assuming $kr$ >> 1, $k$ being the wave number.
- 5.
- Projection of integration on the rough surface on the smooth surface: this projection involves the gradients of the roughness.
- 6.
- Incorporation of a shadow function to approach auto-shadowing: Wagner [74] has proposed to incorporate a shadow function inside the Kirchhoff integral to take into account shadowing effects.

#### 3.2.2. Validity

_{c}cos

^{3}θi >> 1, with r

_{c}the surface local radius of curvature and θi the incident angle. This historical criterion forgets that multiple scattering and shadowing effects are not taken into account in KA leading to large errors for grazing incidences or higher roughness.

^{2}/L of both L and σ, to define a criterion when the KA use is valid. They found that, with a normal incidence angle θi = 0°, KA is valid with a scattering angle −70° < θs < 70°and when σ

^{2}/L ≤ c, the constant upper bound c being in two dimensions 0.20 λp and in three dimensions 0.14 λp (λp being the compressional wavelength). Figure 5 shows a comparison between KA and a finite element (FE) method for a 2D case when σ

^{2}/L~c. A small incidence angle of 30° can improve the KA accuracy when σ

^{2}/L exceeds c as in the case shown in Figure 6. We can however regret that the numerical evaluation of this proposed criterion was limited to small incidences angles: however, this criterion is likely to be suitable for many NDE configurations for which incidence is close to normal. A study of this criterion near grazing incidence should also be useful.

^{2}/σ more than a constant times the wavelength and a condition kL > 6 is also associated. We find the condition on L

^{2}/σ very disbelieving and in contradiction with the previously cited studies and also that given by Voronovich (σ/L < C and large kL; see Figure 1.1 in [20]). The condition kL >6 is also found in other references as [52]. Notably, Figure 2 in [52] represents the stated validity of first-order perturbation theory and KA in kσ–kL space obtained for bistatic scattering cross sections simulation: PT is said valid for small kσ < 2 to 4 depending on kL value, KA valid for kL > 6 and both KA and first-order PT needs σ/L < C (C being a constant value).

^{2}/L); they nevertheless obtained an acceptable disagreement of 2dB (see their Figure 10 [77]) for kL = π/2 (corresponding to L = λp/4) and for an important roughness σ = λp/4 = L equal to the correlation length. It is a pity that that they did not provide numerical validations below kL = π/2.

^{2}/L ≤ Cλp and kL > 1 or 2. For more important observation angles to the normal the criterion on kL and the constant C value are likely to be a little more restrictive: Zhang [85] using 2D FEM calculations recently showed that KA can be considered accurate for kL > π and for incidence and scattering angles over the range from −80° to 80°.

#### 3.2.3. Elastic Targets and Applications

#### 3.2.4. Periodic Surfaces

## 4. More Modern Approaches with Larger Domains of Validity

#### 4.1. Small Slope Approximation (SSA)

#### 4.2. Parabolic Equation (PE)

## 5. Summary, Discussion and Perspectives

^{2}/L ≤ Cλp (C and λp being a constant value and the wavelength) and kL >1 or 2. If we do not want to limit the KA validity to near specular observation, the previous criteria are likely to be a little more restrictive to reach the KA validity for incidence and scattering angles over the range from −80° to 80°(kL > π is for instance needed)

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 1.**Exaggerated examples of rough surfaces generated for CIVA calculations with standard deviation of the height set to 1 mm for different kinds of rough surfaces: (

**a**–

**c**) linear profile for respective ratio $L/\sigma =1$, $L/\sigma =1.5$ (case of Figure 6) and $L/\sigma =4.3$ (case of Figure 11); (

**d**) planar for $L/\sigma =1$; (

**e**) cylindrical for $L/\sigma =1$. Scales in mm.

**Figure 3.**Blakemore’s Figure 8 [36]. Backscattered intensity with the perturbation theory on a steel plate immersed in water. The used parameters are respectively frequency = 2.25 MHz, angular beam width = 3°, observation distance = 300 mm, RMS surface height $\sigma $ = 10 µm and the two correlation lengths of the rough surface both equal to 100 µm. Reproduced with permission from [36], Elsevier, 1993.

**Figure 4.**Blakemore’s Figures 2 and 11 [36]. Experimental validation of Blakemore’s perturbation theory at first order: at left measurements from de Billy et al. [37] on a brass plate immersed in water; at right, Blakemore’s results with frequency = 2.2 MHz, angular beam width = 3°, observation distance = 300 mm, RMS surface height $\sigma $ = 14.9 μm and the correlation lengths ${L}_{x}^{}=90$ μm and ${L}_{y}^{}=105$ μm. Reproduced with permission from [36], Elsevier, 1993.

**Figure 6.**Shi’s Figure 12 [77]. (

**a**) Shi’s Figure 11 [77]. Scattered amplitude versus observation angle using 2D FE and KA simulations, for θi = 30°, σ = λp/3, L = λp/2 < 0.6 λp. (

**b**) Error between 2D FE and KA with respect to θi in the specular direction for different values of σ. Reproduced with permission from [77], Royal Society, 2015.

**Figure 7.**Ultrasonic inspection of a planar steel component (orange box) immersed in water (water path 25 mm) at frequency = 2 MHz (wavelength λ = 0.75 mm): Bscans simulated with KA for a (

**a**) smooth and (

**b**) a rough entry surface (RMS surface height $\sigma $ = 1 mm and correlation lengths ${L}_{x}^{}={L}_{y}^{}=10$ mm). Scattering patterns (maximum of the echographic signal versus observation angle) (

**c**) for different roughnesses (smooth, 10 µm, 0.1, 0.5 and 1 mm) and (

**d**) for three different generated rough surfaces of the same statistical properties (roughness $\sigma $ = 0.2 mm and correlation lengths ${L}_{x}^{}={L}_{y}^{}=10$ mm).

**Figure 9.**Thorsos’ Figure 2 [112]. Comparison of the second-order SSA (2), third-order SSA(3), and fourth-order SSA(4) small slope approximation and Monte Carlo integral (IE) scattering strengths (=10 log of the scattering cross section) for a modest rms surface slope angle. Here, k $\sigma $ = 0.5, kL = 4.0, the RMS slope angle $\gamma $ is 10°, and the incident angle ${\theta}_{i}$ is 45°. Reproduced with permission from [112], AIP Publishing, 1997.

**Figure 12.**Surface topography of a specimen manufactured by wire arc additive manufacturing (WAAM). Reproduced from [141] (Open access, no copyright).

**Figure 13.**Surface topography of a specimen manufactured by selective laser melting (SLM). Reproduced from [137] (Open access, no copyright).

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Darmon, M.; Dorval, V.; Baqué, F.
Acoustic Scattering Models from Rough Surfaces: A Brief Review and Recent Advances. *Appl. Sci.* **2020**, *10*, 8305.
https://doi.org/10.3390/app10228305

**AMA Style**

Darmon M, Dorval V, Baqué F.
Acoustic Scattering Models from Rough Surfaces: A Brief Review and Recent Advances. *Applied Sciences*. 2020; 10(22):8305.
https://doi.org/10.3390/app10228305

**Chicago/Turabian Style**

Darmon, Michel, Vincent Dorval, and François Baqué.
2020. "Acoustic Scattering Models from Rough Surfaces: A Brief Review and Recent Advances" *Applied Sciences* 10, no. 22: 8305.
https://doi.org/10.3390/app10228305