# Acoustic Scattering Models from Rough Surfaces: A Brief Review and Recent Advances
## 5 Summary, Discussion and Perspectives  
A review of acoustic wave scattering from rough surfaces has been done. Several analytical models proposed in the literature have been described in more detail as well as their validity. We are also focused in specific acoustic applications as scattering from ocean, non-destruction evaluation, medical sciences. Two classical approximations have been early devised: the perturbation theory and the Kirchhoff approximation. The perturbation theory (PT) consists in expressing the scattered field in an expansion series using as small parameter the roughness height. This method leads to a major inconvenient: it is valid for low roughness when compared to acoustic wavelength (at first order for small \(k\sigma<2\) to 4 depending on \(kl\) value, \(\sigma\), \(L\) and \(k\) being respectively the rms roughness height, the correlation height and the wave number). Existing validations also showed that PT is rather accurate far from the specular direction. Consequently, it appears possibly more restrictive than the Kirchhoff method for numerous applications notably non-destruction evaluation (NDE) which usually employs inspection in specular direction. The Kirchhoff approximation (KA) leads to a simple analytical expression; it is a relatively powerful and inexpensive approximation in terms of calculations and easily extended to impedance rough surfaces. One important fact is that KA is the most used method in NDE for rough surfaces: its implementation of the rough structure echo consists in simply modifying that of the smooth surface; the \"rough\" KA algorithm can thus simply mesh the smooth mean surface (surface without roughness) in small surface elements and then modify the contribution of each mesh point by including some local geometrical parameters of the rough surface. KA has the advantage of handling surfaces of roughness smaller than a constant depending on the wavelength and on the correlation length with a relatively satisfying validity except at the vicinity of grazing observation and critical angles. Indeed we have connected and merged recent studies of the KA validity and conclude that for many applications using rather near specular observation and nongrazing incident angles, two criteria can be used for KA validity: \(\sigma^{2}/L\leq C\lambda\)p (C and \(\lambda\)p 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 \(-80^{\circ}\) to \(80^{\circ}\)(kL \(>\pi\) is for instance needed)  
It is well admitted that KA validity for application to rough surfaces decreases for grazing angles and the region of inaccuracy in terms of scattering angles is much more important for a fluid/solid interface. One has to pay attention at the KA validity in the vicinity of particular angles (Rayleigh and critical ones). In NDE inspections, these angles are often avoided by the inspector; nevertheless it may be possible to meet them in some geometrical configurations. Improving KA or analytical approximations to better take into account head waves occurring at critical angles is an important challenge already for planar surfaces [128]. Modeling head waves on irregular surfaces is even more complicated and seems intractable using analytical methods [90; 129]. Finite elements methods [130] can be useful to model both the head waves generated at critical angles, the bulk waves radiated near the surface and their possible interferences [90; 129]. For planar surfaces, some smoothing modifications of KA can be proposed [69] in the vicinity of critical angles for better validity, but they do not simulate head waves. A modification of KA using an alternative integral formula for the Kirchhoff integral has just been developed [89], and it could be extended to scattering from a rough surface.