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The phenomenological theory of multi-mode surface wave propagation is applied to a plane structure having a multi-mode discontinuity in impedance. The resulting boundary-value problem is reduced to the solution of a Wiener-Hopf equation whose factorization is given in terms of the factorization that occurred in the one-mode case. Despite the complexity of the solution, the magnitudes of the surface wave excitation coefficients are elementary functions, as is the cylindrical power flow. On computing the power flow through the impedance surface, a definition of subsurface power flow "inside" the structure is suggested. The form can be taken so that the concept of modal power flow separability is maintained wherever the associated exterior field is primarily that of surface waves. It is further observed that without consideration of this term, power flow coupling occurs. Analogous results appear in the exact case of a dielectric slab having multiple simultaneously propagating surface wave modes. Lastly, conservation of power is verified by actual evaluation of the closed contour integral used to define the various components of power flow (surface wave, cylindrical, junction, and boundary). In fact, the following physical interpretation can be made for the magnitudes of the respective power flow distributions: the incident surface wave power (including the associated subsurface power) is equal to the excited surface wave power (including the associated subsurface power) plus the cylindrically radiating power. Introduction. Most exact problems of electromagnetic wave propagation are intractable. As a result, numerous approximate methods have been devised to discuss various aspects of the phenomena. In certain problems involving surface waves, it is known that this feature may be investigated by replacing the details of the structure with an impedance boundary condition. The class of problems to which we direct ourselves is characterized by discontinuous plane structures (infinite in extent) having the capability of supporting several surface wave modes. The orientation and geometry are such that the electromagnetic field produced is determined by solving a two-dimensional problem and all field components are derivable from a single unknown scalar wave function u(x, y). The plane structure will be replaced by the boundary conditions put forth by Karp * Received September 18, 1971. The research reported in this paper was supported partially by St. John's University and partially by the office of Naval Research under Contract No. N00014-67A0467-0075. Reproduction in whole or part permitted for any use of the U. S. Government. 300 R. C. MORGAN AND S. N. KARP and Karal [1], [2], These conditions have the form of generalizing the classical impedance condition to products of this type. Furthermore, they are derivable by approximating the reflection coefficient due to a plane incident wave (which may be known theoretically or experimentally). Previously, these conditions were applied to continuous structures in the papers [2], [3] and [4]. A discontinuity in the boundary condition for a two-mode problem on a right-angled wedge was treated in [5]. This paper will be directed toward solving the resulting mathematical problem of a two-mode discontinuity on a plane structure. A similar single-mode discontinuity problem was discussed by Kay [6]. It is interesting that despite the complexity of the solution, the physically important magnitude of the surface wave excitation coefficients is easily obtained (Kay observed this in the single-mode case). We take as our incident field an appropriate surface wave, and for simplicity, one half of the structure is allowed to be slightly thick (see [4]) while the other half is perfectly conducting. The various components of the power flow above the structure are computable by choosing an appropriate contour in physical space. In particular, on computing the power flow through the impedance surface, we are led to a definition of power flow inside and down the structure. The form is such that the concept of modal power flow separability is maintained wherever the associated exterior field is primarily that of surface waves. It is well known that, in the exactly solvable case of an infinite dielectric slab having simultaneously several propagating surface wave modes, surface wave modal power flow separability results if the power flow across a plane perpendicular to the slab is computed by including the associated power flow inside. Otherwise, there is in general a coupling of power when it is computed only above the slab. This phenomenon is observed also in our impedance model. For the case of the single real impedance, a consideration of this type does not occur because power is not transferred across the impedance surface and the model is one for which the power flowing inside the structure is negligible. In the case of the dielectric slab mentioned above, this corresponds to a slab of vanishing thickness. However, for a slab of finite thickness, the power flow inside may be significant. Thus, by our model, we provide an approximation that accounts for such power flow while the structure is also discontinuous. The exact problem of such a discontinuous structure has not been solved. However, we point out that the phenomenological theory is broader in context than the problem of the dielectric slab. Thus, we conjecture that our results herein are also. As a partial verification of our computations, conservation of power is checked for the contour used to define the various component field power distributions (i.e., surface wave, cylindrical and boundary). It is shown that the surface wave power (including that amount associated with flow inside the boundary) exactly cancels out the cylindrically radiated flow. The remaining boundary terms give zero power flow. As a result, we can state the following physical theorem for the magnitudes of the power flow distributions: the incident surface wave power flow (including "inside" term) is equal to the excited surface wave power flow (including "inside" term) plus the cylindrically radiated power flow. The boundary-value problem formulation is modeled after that used previously in proving uniqueness for a multi-mode right-angled wedge problem [11]. In addition, the junction condition assumed arises naturally upon reducing the problem to the solution of a Wiener-Hopf equation. Here the factorization is given in terms of the factorization that occurred in the one-mode case. SURFACE WAVE INCIDENCE ON A PLANE STRUCTURE 301 1. B.V.P. formulation and solution. The problem (see Fig. 1) we shall solve is posed by the following conditions: (i) (A + k)u{x, y) = 0, all x, y > 0 (ii) (d/dy + Xi)(d/dy + X2)w = 0, x < 0, y = 0, where Xi , X2 are taken as real positive constants. (iii) u = 0, x > 0, y = 0. (iv) u and its derivatives satisfy d'u (a) X) Z dx' ' dy' < M for r > R0 , where M is independent of r and 8 and R0 is some positive constant, (b) d2u/dy2 is integrable at the origin. (v) 11 ^incident *~i~ Wexcjted ~i~ Wradiftted winc. = A exp I — \^y + i(k2 + X?)1/2a:], x < 0, y > 0, = 0 , x > 0, y > 0, and 2 w«=ited = X) Cm exp [-X„2/ i{k2 + X2)1/2z], x < 0, y > 0, m -1 = 0 , x > 0, y > 0. Here A represents the given incident surface wave amplitude and the C„ are constants that must be determined. (vi) wauled = u — wiD0. — uCTc,ted obeys the radiation condition lim \/r (dWrad.M" — ikurlli) = 0 r—*co uniformly in 6, 0 < 6 < tr. Following a method employed by Kane and Karp [7] and Kane [8], the solution is reduced to solving a Wiener-Hopf problem. The critical factorization required is that of the function x, , x.) f[ [l + ^ i->y,] (l.l)

Surface wave incidence on a plane structure having a multi-mode discontinuity in impedance