# Acoustic Scattering Models from Rough Surfaces: A Brief Review and Recent Advances
## 3 Historical Approximations
### The Perturbation Theory  
The perturbation theory (PT) assumes some restrictive hypotheses on the local height \(h\) and gradient \(\
abla h\) at points of a rough surface:  
\[k|h| <<1 \tag{8}\] \[|\
abla h| <<1 \tag{9}\]  
\(k\) being the incident wavevector modulus. It is assumed that if previous conditions are satisfied, the roughness is weak compared to acoustic wavelength and slowly varies. It expresses the scattered field at an observation point \(r\) as an expansion series:  
\[\psi(r)=\psi^{inc}(r)+\sum_{n=0}^{\infty}\psi_{n}^{sc}(r) \tag{10}\]  
where \(\psi^{inc}\) is the incident wave field and \(\psi_{n}^{sc}\) is the nth-order approximation to the scattered field. The \(n\) = 0 term is the scattered field from a smooth surface. Then the main principle of the perturbation Theory consists in assuming that the boundary conditions may be expanded in a Taylor series about the mean plane \(z\) = 0. Mathematically, the boundary conditions on the rough surface expressed as:  
\[f(x,y,z)=0 \tag{11}\]  
where \(f\) can be, for instance, the acoustical pressure for Dirichlet conditions--which can be expanded in a Taylor series about the mean plane \(z\) = 0 as follows:  
\[f(x,y,z)|_{z=h(x,y)}=f(x,y,z)|_{z=0}+h\frac{\partial f}{\partial z}|_{z=0}+ \frac{h^{2}}{2}\frac{\partial^{2}f}{\partial z^{2}}|_{z=0}+\ldots. \tag{12}\]  
Some applications of the perturbation theory consider first scattering from canonical rough surfaces [18]. For instance for the [PERSON]'s problem (scattering from a surface excited by a load force applied at one particular surface point) [34], the solution is expressed as convolution integrals evaluated only for a triangular irregularity of small height compared to its width and to the wavelength. The application of PT to random rough surfaces leads to consider separately a mean coherent contribution and an incoherent one. For the coherent part, reflection and transmission coefficients are determined. For the incoherent part, the mean intensity of the scattered field is calculated [35]. The determined first-order PT term can be substituted back into the Helmholtz scattering formula to iteratively deduce higher order terms. [PERSON] [36] has then presented a Perturbation Theory at the first order for a fluid/solid interface. PT simulations in backscattering for rigid and elastic solids are very close except at the vicinity of the Rayleigh angle as shown in Figure 3.  
Figure 2: Variations of heights around an average plane. In this region a small peak in the intensity of the incoherently scattered field is predicted, consistent with experiment. Nevertheless, in the case of the proposed experimental validation in pulse echo configuration [37] (Figure 4), [PERSON] prediction (not shown here) are about 3 dB lower than PT one at backscattering angles theta = 40\({}^{\circ}\), and 5.5 dB lower at theta = 70\({}^{\circ}\). Some applications and references of PT for scattering from fluid/fluid, fluid/solid interfaces and fluid/stratified layers [38; 39; 40] for inhomogeneous media [41; 42] can be found in the literature, generally for ocean applications. An acoustic model in the time domain based on first-order perturbation theory for scattering from layered sediments has recently been proposed [43]. Figure 4: [PERSON]’s Figures 2 and 11 [36]. Experimental validation of [PERSON]’s perturbation theory at first order: at left measurements from [PERSON] et al. [37] on a brass plate immersed in water; at right, [PERSON]’s results with frequency = 2.2 MHz, angular beam width = 3\({}^{\circ}\), observation distance = 300 mm, RMS surface height \(\sigma\) = 14.9 \(\upmu\)m and the correlation lengths \(L_{x}=90\)\(\upmu\)m and \(L_{y}=105\)\(\upmu\)m. Reproduced with permission from [36], Elsevier, 1993. Figure 3: [PERSON]’s Figure 8 [36]. Backscattered intensity with the perturbation theory on a steel plate immersed in water.