In 1963 Uretsky described the reflection of plane waves from a sinusoidal boundary as a ’marvelously complex problem’ [36]. This complexity has lead to many different models being proposed; what follows is a brief review of the literature (see for exam-ple [37–41] for more information). All scattering models are essentially attempting to solve the Helmholtz scattering integral over a closed surface S, which gives the total field ψ at a point r′ away from the surface [42].
ψ (r) = ψinc(r′) + ∫
S
(G (∣r′−r∣)∂ψ (r)
∂n −ψ (r)∂G (∣r′−r∣)
∂n )dS (2.12)
where n is the unit inner normal at the surface, r is a point on the surface, G (∣r′−r∣) is the free space Green’s function and the second term in the equation represents the scattered field ψsct. Figure 2.2 illustrates the problem geometry. The method of solving equation 2.12 and the choice of Green’s function depends on the shape of the surface and the application of the theory. Much of the early literature dis-cussing the scattering of waves by periodic boundaries was concerned with reflection
2. Basic Principles of Bulk Waves and Scattering
n r
r
0ψ
incS
x y
ψ
sctFigure 2.2: Problem geometry of wave scattering by a rough surface.
of acoustic waves from the surface of the sea which was assumed instantaneously sinusoidal. In 1878 Rayleigh proposed that the scattered wave from such a surface could be expressed as a sum of infinite plane waves travelling away from each point on the surface [43]. His assumption that this would hold at any point above the surface was heavily criticized making it valid only for small amplitude undulations;
his initial method was subsequently altered by many. Marsh for instance gener-alised the theory and extended application to rough surfaces [44]. Uretsky sought to abolish both assumptions completely and in doing so created a mathematically more rigorous yet complex analytical solution for sinusoidal pressure release surfaces showing satisfactory agreement with experimental results performed by Barnard et al [36, 45].
Today, solutions to the problem of rough surface scattering can be broadly split into two main categories; those which use approximations to simplify analytical scattering models and those requiring no approximation which include numerical and boundary integral equation (BIE) methods. In the absence of sufficient com-putational capabilities, the majority of early literature focussed on approximate analytical solutions which aimed to reduce complexity, all be it at the cost of accu-racy. Elfouhaily and Gu´erin [39] estimate that over twenty different models exist in literature with little information pertaining to the conditions under which each can be applied and assumed valid. Kirchhoff theory, which is one of the most commonly implemented, approximates the derivative of the wavefield along the boundary by assuming that at each point the reflecting surface acts as an infinite plane reflector with an orientation equal to that of the boundary. Splitting scattered field con-tributions up in this manner represent the main drawback of this and many other
2. Basic Principles of Bulk Waves and Scattering
approximate analytical solutions to the scattering integral: any components which have interacted with the boundary multiple times are not calculated. Without al-teration phenomena such as surface self-shadowing and diffraction are also generally not well modeled using such approaches, limiting the levels of surface roughness which can be investigated. Information regarding the validity of the Kirchhoff ap-proximation can be found in [46, 47].
Numerical techniques such as the finite element method (FEM) require no assump-tions to be made about the field along the reflecting boundary, making the solution exact within stability limits representing their main advantage over approximate analytical techniques. Field variables are calculated at many nodal points making up a mesh which discretises the entire region of interest, often including absorbing boundaries to damp out any unwanted reflections. Zhang et al [48] apply the efficient frequency domain FEM model described by Velitchko et al [49] to the calculation of the far field scattering matrix of longitudinal waves from rough defects and com-paring results when using the Kirchhoff approximation. Constructing the mesh only within the near field of the defect and using absorbing boundaries in the method de-scribed improves efficiency over the standard FEM; in general however, any method requiring mesh generation around defects still requires greater computational effort than those which calculate field variables only along the boundaries. For three di-mensional scattering these computational requirements are particularly apparent, making numerical methods unsuitable for some of the investigations carried out in this thesis.
The final approach to solving equation 2.12 involves formulating a BIE based on the boundary conditions on a rough surface between two homogeneous media which can either be solved directly or by optimizing the solution from a set of fictitious point sources [41]. Direct solution of the analytical expression at a rough boundary is incredibly complex and as such no widely implemented solutions exist. DeSanto [50] presents a BIE for scalar wave scattering from infinite rough surfaces using the analytical expressions in their exact form, finding that the solution reduces down to that expected from a planar reflector at the flat surface limit. However, for Dirichelet boundary conditions it was found that a hypersingular integral equation results, making the algebra far more complex. Specifying an incident plane wave also
2. Basic Principles of Bulk Waves and Scattering
makes the approach unsuitable for the investigations carried out in this thesis. A method was thus developed using the distributed point source method (DPSM) [51]
which solves the BIE through semi-analytical means by specifying fictitious point sources which are in close proximity to the boundaries.
Initially proposed by Placko and Kundu in 2000, the DPSM is a semi-analytical simulation technique which has been developed to model magnetic, electrostatic, electromagnetic and ultrasonic problems [51]. It operates by placing point sources within close proximity to boundaries; by using the known boundary conditions the complex amplitudes of these point sources can be calculated which are subsequently used to propagate the wavefield to any set of target points within the problem do-main. Using the analytical solution of a point source directly in this manner avoids meshing, improving computational efficiency when compared with fully meshed nu-merical techniques, particularly when considering large three dimensional problem geometries or long time scales. To the authors knowledge the technique has never previously been applied to the problem of rough surface scattering, an application which has been identified through work presented in this thesis as benefitting greatly from the accuracy and efficiency offered by the technique. More information regard-ing its development by others can be found in section 3.2