#### 2.1. Essential Definitions

[6] Let us first consider a model initial boundary-value problem for a certain type of discontinuity in waveguide of arbitrary cross section. (see Figure 1a). Formally this 3D scalar problem corresponds to the propagation of acoustical waves. The results derived in course of its solving can be easily extended for the case of electromagnetic waves propagation. We have chosen it as a model problem only because the 3D vector electromagnetic problems are quite cumbersome for the description of rather general principal questions. In the same time the description of 2D scalar problems would restrict the class of problems that are planned for consideration significantly.

[7] Function *U* (*g*, *t*) describing wave scattering process is subject to the equations

We assume that the finite in the domain functions φ(*g*), *F*(*g*, *t*), Ψ(*g*), ε(*g*) − 1 and σ(*g*) satisfy the conditions of the theorem about single-valued solvability of (1) in the energy space (Sololev's space) **W**_{2}^{1} (**Q**^{T}), **Q**^{T} = **Q** × (0, *T*), *T* < ∞ [see *Ladyzhenskaya*, 1985]. Here *L* is the differential elliptical operator of second order that takes in the local cylindrical coordinates *x*_{j}, *y*_{j}, *z*_{j} the form ∂^{2}/∂*z*_{j}^{2} + ; *M* is the differential operator not higher than of the first order; **S**_{z} is the boundary of domain **Q** that is constituted by two semi-infinite regular (by *z*_{j} > 0, *j* = 1, 2) waveguides that are bound by a compact discontinuity; *g* = {*x*, *y*, *z*} ; time *t* has the length dimension; the real functions ε(*g*) ≥ 1 and σ(*g*) ≥ 0 determine the influence of the discontinuity on the excitation propagation velocity and its dissipative characteristics.

[8] In a particular case of an infinite regular waveguide (section int **S** of the domain **Q** by the planes *z*_{j} = const is constant along the axis *z*, and ε(*g*) − 1 = σ(*g*) ≡ 0), let us write the solution to (1) like

where the sequence of functions *v*(*z*, *t*) = {*v*_{n} (*z*, *t*)} satisfy the equations

and the initial conditions

and {μ_{n} (ρ, ϕ)} and {λ_{n}} are sets of the eigen functions and eigen values of the homogeneous boundary-value problem

that is derived from (1) by separating the cross coordinates *x*, *y* (**S** is the boundary of the domain int*S* and the supposed properties of operators *L*_{x, y} and *M* are so that the function system {μ_{n} (*x*, *y*)} constitute an orthonormal basis in the corresponding plane domain). Here *a*_{n} (*z*, *t*), *b*_{n} (*z*), *c*_{n} (*z*) are the Fourier coefficients of functions *F* (*g*, *t*), φ (*g*), Ψ(*g*), by expanding the latter into series over the basis system {μ_{n} (*x*, *y*)}; and {*n*} is an ordered set of numbers *n*.

[9] Going to the generalized definition of the Cauchy problems (3), (4), see [*Vladimirov*, 1971] and using the fundamental solution *G*(λ; *z*, *t*) = (−1/2) χ (*t* − ∣*z*∣) *J*_{0} [λ (*t*^{2} − *z*^{2})^{1/2}] of operator *D*(λ) (by definition *D*(λ)[*G*(λ)] = δ(*z*)δ(*t*)), we yield

Here the asterisk stands for the convolution operation, *f*_{n} (*z*, *t*) = *a*_{n}(*z*, *t*) − δ^{(1)} (*t*)*b*_{n}(*z*) − δ(*t*)*c*_{n}(*z*), χ is the Heaviside function, *J*_{m}—Bessel function, δ^{(m)} is the generalized derivative of the delta-function of the order *m*. Equations (2) and (5) demonstrate the general and specific, corresponding to the given sources, form of wave propagating in a regular waveguide. Its alteration in space and time are determined exclusively by the system *v* (*z*, *t*) (or {*G* (λ_{n}) * *f*_{n}}) whose properties allow it to serve as a universal evolutionary basis of any signal on any finite section of a regular guiding structure.

[10] Let the wave of the type (2) serve as the excitation wave to the open waveguide resonator shown in Figure 1a. The excitation field assumed to be non-zero only in the waveguide *A* that is regular for all *z*_{1} > 0. The field reaches the left boundary of the discontinuity lying in the plane *z*_{1} = 0 after the time *t* = 0. The secondary field that is induced in the regular semi-infinite waveguides *A* and *B* and propagates toward the increasing values of *z*_{1} and *z*_{2} we present like

Through the relations

we introduce the boundary (at the boundaries of the discontinuity *z*_{j} = 0; δ_{m}^{n} is the Kronecker delta) transform operators *R*^{AA} and *T*^{BA} of the evolutionary basis of the non harmonic wave arriving from the left, from the waveguide *A*:

The entries *R*_{nm}^{AA} (*t* − τ) and *T*_{nm}^{BA} (*t* − τ) of these operators define the space-time distribution of the incident wave energy *U*^{i} (*g*, *t*) in the reflected and the passed through the waveguide resonator field. The connection of the domains is defined in following way: the upper index from the domain whose identifier stands on the right to the domain with the identifier on the left. The connection of the modes: inferior index; on the right is the mode number of the incident wave, on the left is the mode number of the secondary field.

[11] Obvious that *R*^{AA} and *T*^{BA} operating in the domain of the signal's evolutionary basis are the characteristics of the waveguide unit itself and just sum up properly the results of a multiple elementary disturbances that can constitute any primary signal *U*^{i} (*g*, *t*). Thus, if *v*_{n1} (0, *t*) = δ_{n}^{p} δ (*t* − η), where *p* is an integer and η > 0, we have *w*_{n1}′ (0, *t*) = *R*_{np}^{AA} (*t* − η) and *w*_{n2}′ (0, t) = *T*_{np}^{BA} (t − η). The use of such an abstract, physically impossible signal is justified by the methodological backgrounds, namely through this signal the structure gets elementary “excited” that enables one to extract from the generated field the “pure” components *R*_{np}^{AA} (*t* − τ) and *T*_{np}^{BA} (*t* − τ) of the transform operators. The elements of the matrix-functions *R*^{AA} and *T*^{BA} can be certainly computed in a different way by searching the sequence of values *p* in the excitations waves with *U*^{i} (*g*, *t*), with *v*_{n1} (*z*_{1}, *t*) = δ_{n}^{p}*v*_{n1} (*z*_{1}, *t*), *n* ∈ {*n*(1)}, where *v*_{n1} (*z*_{1}, *t*) are now sufficiently arbitrary functions. From (7) follows the relation between the amplitudes *w*_{nj}′(0, *t*) of the secondary field corresponding to the fixed value of *p* and the sought values:

After its inversion using the operation method we obtain

where *L* and *L*^{−1} are the direct and inverse Laplace transforms. There are many ways to implement the corresponding presentation with the minimal error and acceptable computational efforts. Properly chosen function *v*_{p1} (0, *t*) is a crucial factor here. In an ideal case *v*_{p1} (0, *t*) has to have such a form that the necessary integral transformations could be done analytically.

[12] The operators *R*^{AA} and *T*^{BA} are the boundary ones, they determine all the characteristics of the transient processes directly at the discontinuity boundary of a regular guiding structure. The secondary field, departing from this boundary, propagates “freely” in the semi-infinite regular channels and in course of it becomes deformed. The space-time amplitudes of the modes {*w*_{nj} (*z*_{j}, *t*)} (signal's evolutionary basis) varies in time and distance in different way for various values of *n* and *j*. These changes can be described by using the diagonal transport operators *Z*^{A} (*z*_{1}) and *Z*^{B} (*z*_{2}) that are subject to the rule

The structure of these operators is detailed by the formula

that represent the common feature of the solution to the homogeneous equations of the type (3) at the semi-axes *z*_{j} ≥ 0. The solutions that satisfy the zero initial conditions and are free from any components propagating along the decreasing *z*_{j}. This formula is derived through the integral Laplace transform over *t* or the Fourier cosine transform over *z*_{j} [*Maikov et al.*, 1986]. Actually, on the assumption that at the initial moment of time *t* = 0 the excitation wave *U*^{j} (*g*, *t*) has not yet reached the discontinuity boundary in the plane *z*_{1} = 0, we obtain for the amplitudes {*w*_{nj} (*z*_{j}, *t*)} a sequence of homogeneous initial boundary value problems:

By using in (11) the Fourier cosine transform over *z*_{j} at the semi-axes *z*_{j} ≥ 0 (image original)

we obtain the following Caushy problems for the images _{nj} (ω, *t*):

Here we had taken into account that the waves *U*_{j}^{s} (*z*_{j}, *t*) are free of components propagating along the decreasing *z*_{j}. The components traveling toward *z*_{j} = ∞ at any finite moment of time *t* = *T* are zero for sufficiently large *z*_{j}. The convolution of the fundamental solution (λ; *t*) = χ (*t*)λ^{−1} sin λ*t* of operator (λ) with the right-hand part of (12) enables us to write _{nj} (ω, t) in the form

Formula (10) that defines the diagonal transport operators describing the field variation by a “free” propagation of the wave *U*_{j}^{s} (*g*, *t*) in the finite sections of the regular waveguides *A* and *B* we obtain after transforming (13) to the originals *w*_{nj}(*z*_{j}, *t*)).

#### 2.2. The Method of Transform Operators For Evolutionary Basis of Signal

[13] The above-introduced operators describe on the whole the scattering properties of discontinuities that are being excited from the channel *A*. Let us determine now, similarly to *R*^{AA} and *T*^{BA}, the transformation operators *R*^{BB} and *T*^{AB} of the evolutionary basis of the wave arriving from the right, through the waveguide *B*, at the boundary *z*_{2} = 0:

or in operator form

Assuming the sets of operators *R* and *T* to be known for separate simple discontinuities, we develop an algorithm for analyzing the scattering properties of a compound unit constituted by such discontinuities. In the situation modeled that is shown in Figure 1b, the unit contains two sequent discontinuities **I** and **II** connected by a section *B* of the regular waveguide of a finite length *L*, and is excited by a wave of the type (2) arriving at the boundary *z*_{1}(*I*) = 0 from the waveguide *A*. Remaining in the frameworks of the above accepted terms, (the obvious changes are due to two different types of discontinuities **I** and **II**), we present the solution of the corresponding initial boundary value problem in the regular domains *A*, *B* and *C* in the form

The first group of components here corresponds to the waves propagating from the left to the right, the second groups - to the wave moving from the right to the left (see Figure 1b). Following the expressions (7)–(10), (14), and (15), and using the denotations

we obtain the following set of operator equations:

This set clearly demonstrates all the stages of forming a feedback of a complex structure to the excitation by the signal with the evolutionary basis *v*_{1}(*z*_{1}(**I**), *t*) = {*v*_{n1}(*z*_{1}(**I**), *t*)}, or simply the signal ν_{1}(**I**). Thus, e.g. the first equation of the set is a sum of signals first of which appears as a result of the reflection of the primary signal ν_{1}(**I**) from the discontinuity **I**, and the second one is produced by the signal *w*_{1}(**II**) that has passed through this discontinuity and the regular space *B*.

[14] By eliminating from (16) one unknown function, we arrive to an operator equation of the second kind with respect to the unknown vector-function *w*_{2}′(**I**)

and to the “recalculating” formulas determining all the components of the field generated by the unit. The operator in the right-hand part of (17) influences the sought vector-functions *w*_{2}′(**I**) whose argument τ is strictly less than the time *t* in the argument of the same function standing on the left (the lag effect that is conditioned by the finite velocity of the signal propagation). Hence, the numerical solution of the final equation can be obtained within the standard scheme of marching over time layers, thus, the initial “complicated” problem is equivalently re-formulated to the form allowing the direct inversion by using the conventional methods of computational mathematics. The complex unit is reduced to the category of elementary basic blocks after calculating the elements of boundary operators by the formulas (7) and (14).

[15] Let us return once more to the expressions (7)–(10), (14), and (15) to prevent the eventual questions concerning the definition of boundary transformation operators. The point is that these operator act in a different way if compared with their analogues in the frequency domain. Theoretically, one can choose the traditional way in the time domain establishing relations by the boundary transformation operators, the pair “field field” and not the pair “field the field derivative toward the propagation”, as it was done in (7) and (14). Naturally, in such case we would have obtained *R*_{pp}^{AA} (*t* − τ) = −δ (*t* − τ) instead of, e.g., quite a complicated expression [see *Shestopalov et al.*, 1986a, 1986b] describing the characteristics of the simplest discontinuity “waveguide end cap”. Let us consider now more detailed the structure of the transport operators Z^{A}(*z*_{1}) and Z^{B}(*z*_{2}) (formulas (9) and (10)). It was the optimal, in terms of the computational costs, scheme of including them into the algorithm of analyzing the complex unit (see (16) and (17)), that predetermined the quite not physical choice made in (7) and (14). The transformation “field field” is implemented by the product of the operators *ZR* and *ZT* directly, without any intermediate differentiating operator toward the signal propagation direction.