Asymptotic theory for inhomogeneous waves

Abstract

The conventional asymptotic theory for propagation of high-frequency fields is based on a local description in terms of homogeneous plane waves <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">A(r) exp [ik_{0}S(r)]</tex> , where <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k_{0}</tex> is the (large) free space wavenumber, <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">A(r)</tex> is a spatially dependent amplitude, and the phase <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">S(r)</tex> is real. The conventional theory does not accommodate the more general class of fields that behave locally like inhomogeneous plane waves with complex phase <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">S(r)= R(r) + iI(r)</tex> , where <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">R</tex> determines the propagation of the equiphase surfaces and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">I</tex> describes the attenuation. This paper develops an asymptotic theory for inhomogeneous wave fields in lossless media, to be termed evanescent fields. Such fields are encountered, for example, in connection with Gaussian beams and with phenomena on the exterior of surface wave structures or on the dark side of caustics. The scalar wave equation is used to derive eikonal and transport equations for <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">S</tex> and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">A</tex> , respectively, and it is shown how the latter equations may be integrated with the aid of trajectories tangent to the direction of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\nablaR</tex> , which differs slightly from that for the local power flow. Detailed application of the theory is made to two-dimensional scattering of a weakly evanescent incident plane wave by a curved boundary in a homogeneous medium. The phase propagation paths for the reflected field are determined explicitly and are found to possess curvature and points of inflection; these characteristics are shown to be predictable from basic attributes of evanescent wave propagation. For the special case of a circular cylinder, the subsequently constructed reflected field is found to agree with the asymptotic expansion of the rigorous solution, thereby confirming the validity of the theory for weakly evanescent fields. The rigorous solution, valid for arbitrary evanescent decay and obtained from known results for ordinary plane wave scattering by analytic continuation of the incidence angle to complex values, reveals that both the reflected and creeping wave fields should be viewed in a restrictive manner when the evanescent decay is large. However, for weak evanescent decay, these fields retain their customary significance and permit their construction by local evanescent field tracking. It is interesting to observe that in contrast to the nonevanescent case, the creeping waves provide a field contribution exceeding that of the incident or reflected waves in certain portions of the illuminated region.