We assume that the problems of interest are well-approximated by the inhomogeneous Helmholtz equation
[^2 + k_0^2n^2(x)]u(x) = s(x), where is the wave amplitude, is a localized source function, is the wavenumber of the homogeneous background, with being angular frequency, f frequency, c0 the assumed homogeneous background wave speed, and wavelength. The acoustic index of refraction is
n(x) = c_0c(x), where is the wave speed at spatial location .Thus, in the background and measures the change in the wave speed at the scatterers.
Pertinent fundamental solutions for this problem satisfy:
[^2 + k_0^2]G_0(x,x') = -(x - x') and
[^2 + k_0^2n^2(x)]G(x,x') = -(x - x'), for the homogeneous and inhomogeneous media, respectively. In both cases, we assume the radiation (out-going) boundary condition at infinity.
The well-known solution of (g0) for the homogeneous medium in 3D is
G_0(x,x') = e^ik_0x - x' 4x - x'. The fundamental solution of (g) for the inhomogeneous medium can be written in terms of that for the homogeneous one (in the usual way) as
G(x,x') = G_0(x,x') + k_0^2a(y)G_0(x,y')G(y,x')d^3y. Note that the right hand side depends on the values of the G, which is to be determined by the same equation. So this is an implicit integral equation that must be solved for .The regions of nonzero are assumed to be finite in number (N), in compact domains , all of which are small compared to the wavelength . Then, there will be some position (certainly for convex domains) inside each domain ,characterizing the location of each of the N scatterers. We also call these scatterers ``targets,'' since it is their locations that we seek.
With these assumptions, it is a good approximation to set the arguments of G0 and G inside the integral equal to for all 's inside domain . Then, the fundamental solutions can be moved outside of the integral. There remains the integral over , incorporated into the scattering coefficient
q_n k_0^2__n a(y)d^3y, which then characterizes the strength of the scattering from the nth target domain.
With these definitions, we finally have
G(x,x') G_0(x,x') + _n=1^N q_n G_0(x,y_n)G(y_n,x'). Furthermore, if the scatterers are sufficiently far apart and the scattering strengths qn are not too large, then on the far right can be replaced by , giving the explicit formula
G(x,x') G_0(x,x') + _n=1^N q_n G_0(x,y_n)G_0(y_n,x'). Equation (Born) is the Born approximation to for small scatterers.