Generalization of the Maggi-Rubinowicz Theory of the Boundary Diffraction Wave—Part I†

As a first step towards a generalization of the Maggi-Rubinowicz theory of the boundary diffraction wave, a new vector potential W (Q, P) is associated with any monochromatic scalar wavefield U (P). This potential has the property that the normal component of its curl, taken with respect to the coordinates of any point Q on a closed surface S surrounding a field point P, is equal to the integrand of the Helmholtz-Kirchhoff integral; that is, curlQW (Q, P) ·n=14πU (Q) ∂∂n (exp (iks) s) −exp (iks) s∂∂nU (Q). where s is the distance QP and ∂/∂n denotes the differentiation along the inward unit normal n to S. Further it is shown that the vector potential always has singularities at some points Q on S and that the field at P may be rigorously expressed as the sum of disturbances propagated from these points alone. A closed expression for the vector potential associated with any given monochromatic wavefield that obeys the Sommerfeld radiation condition at infinity is derived and it is shown that in the special case when U is a spherical wave, this expression reduces to that found by G. A. Maggi and A. Rubinowicz in their researches on the boundary diffraction wave.