Multi-mode surface wave phenomena

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This paper illustrates, by means of an explicitly solvable problem, use of a slightly singular solution to demonstrate certain aspects of surface wave propagation. The solution is required to be bounded. However, one of the first derivatives is allowed to be singular at a junction of the structure. The surface wave guidance action of the structure is modelled by a two-mode phenomenological boundary condition (multiple impedance condition). For simplicity of illustration, a plane structure that is terminated by a perfectly conducting plane vertical to it, is selected. The solution to the resulting mathematically formulated two-dimensional quarter plane problem is obtained exactly in an elementary fashion. Thus, it is possible to simply observe certain properties of surface wave propagation. The main result is that the “sub-surface” power flow at the junction depends directly on the amplitude of the slightly singular solution, leading us to conjecture that such fields represent leakage or excitation through the junction. Additionally, we show that control over the form of the excited outward travelling surface waves may be exercised by inclusion of this term or by a slot imbedded in the vertical termination, i.e., mode conversion is possible. Finally, by simultaneously adjusting the termination and the slot, the structure can be made to be only cylindrically radiating. The multiple impedance structure is extended in a “virtual” manner so that the effect of its internal field may be taken into account when computing the total power flow in the surface wave field. In particular, modal surface wave power flow is determined by associating an interior field with each component of the exterior surface wave field. The form is suggested by actually calculating the power flow through the impedance surface, a conservation of power argument, and requiring that modal surface wave power flow separability results. After this, by a conservation of energy argument, a definition of power flow everywhere within the virtual structure is given that extends the form previously obtained for the surface wave field. Applying this formula, the flow into the junction is defined. It is easily seen that this power depends directly on the strength of the singular term. Specifically, it vanishes when this term is ruled out by a boundedness requirement. Finally, as an interesting application of a lemma of Lurye and Newstein, the cylindrical power is computed without resorting to complicated integrations of the pattern function and in a simplified case the result is checked by an actual integration.