<p>A method is developed for solving the two dimensional Helmholtz equation in a ductwith varying cross-section region bounded by a curved top and flat bottom, having oneregion inside. To compute the propagation of sound waves in a curved duct with a curvedinternal interface is difficult problem. One method is to transform the wave equation intoa solvable form and making the curved interface plane. To this end a local orthogonaltransformation is developed for the varying cross-sectional duct having one medium inside.This transformation is first used to make the curved top of the waveguide flat andto transform the Helmholtz equation into an initial value problem. Later on the local orthogonaltransformation is developed for a waveguide having two media inside with flattop, a flat bottom and a curved interface. This local orthogonal transformation is used toflatten the interface and also to transform the Helmholtz equation into a simple, solvableordinary differential equation. In this paper we present operator formulation for the partwith flat bottom and curved top including a curved interface. In the ordinary differentialequation with operators in coefficients, obtained after the transformation, all the operationsrelated to the transverse variable are treated as operators while the derivative withrespect to the range variable is kept.</p>

A method is developed for solving the two dimensional Helmholtz equation in a ductwith varying cross-section region bounded by a curved top and flat bottom, having oneregion inside. To compute the propagation of sound waves in a curved duct with a curvedinternal interface is difficult problem. One method is to transform the wave equation intoa solvable form and making the curved interface plane. To this end a local orthogonaltransformation is developed for the varying cross-sectional duct having one medium inside.This transformation is first used to make the curved top of the waveguide flat andto transform the Helmholtz equation into an initial value problem. Later on the local orthogonaltransformation is developed for a waveguide having two media inside with flattop, a flat bottom and a curved interface. This local orthogonal transformation is used toflatten the interface and also to transform the Helmholtz equation into a simple, solvableordinary differential equation. In this paper we present operator formulation for the partwith flat bottom and curved top including a curved interface. In the ordinary differentialequation with operators in coefficients, obtained after the transformation, all the operationsrelated to the transverse variable are treated as operators while the derivative withrespect to the range variable is kept.

Let $l^2(Bbb Z)$ be the Hilbert space of square summable double sequences of complex numbers with standard basis ${e_n:ninBbb Z}$, and let us consider a bounded matrix $A$ on $l^2(Bbb Z)$ satisfying the following system of equations egin{itemize} item[1.] $lan Ae_{2j},e_{2i} an=p_{ij}+alan Ae_{j},e_i an$; item[2.] $lan Ae_{2j},e_{2i-1} an=q_{ij}+blan Ae_{j},e_{i} an$; item[3.] $lan Ae_{2j-1},e_{2i} an=v_{ij}+clan Ae_{j},e_{i} an$; item[4.] $lan Ae_{2j-1},e_{2i-1} an=w_{ij}+dlan Ae_{j},e_{i} an$ end{itemize} for all $i,j$, where $P=(p_{ij})$, $Q=(q_{ij})$, $V=(v_{ij})$, $W=(w_{ij})$ are bounded matrices on $l^2(Bbb Z)$ and $a,b,c,dinBbb C$. This type dyadic recurrent system arises in the study of bounded operators commuting with the slant Toeplitz operators, i.e., the class of operators ${{cal T}_vp:vpin L^infty(Bbb T)}$ satisfying $lan {cal T}_vp e_j,e_i an=c_{2i-j}$, where $c_n$ is the $n$-th Fourier coefficient of $vp$. It is shown in [10] that the solutions of the above system are closely related to the bounded solution $A$ for the operator equation [ phi(A)=S^*AS=lambda A+B, ] where $B$ is fixed, $lambdainBbb C$ and $S$ the shift given by ${cal T}_{arzeta+arxi z}^*$ (with $zetaxi ot=0$ and $|zeta|^2+|xi|^2=1$). In this paper, we shall characterize the ``eigenvectors" for $phi$ for the eigenvalue $lambda$ with $|lambda|leq1$, in terms of dyadic recurrent systems similar to the one above.

Let $l^2(Bbb Z)$ be the Hilbert space of square summable double sequences of complex numbers with standard basis ${e_n:ninBbb Z}$, and let us consider a bounded matrix $A$ on $l^2(Bbb Z)$ satisfying the following system of equations egin{itemize} item[1.] $lan Ae_{2j},e_{2i} an=p_{ij}+alan Ae_{j},e_i an$; item[2.] $lan Ae_{2j},e_{2i-1} an=q_{ij}+blan Ae_{j},e_{i} an$; item[3.] $lan Ae_{2j-1},e_{2i} an=v_{ij}+clan Ae_{j},e_{i} an$; item[4.] $lan Ae_{2j-1},e_{2i-1} an=w_{ij}+dlan Ae_{j},e_{i} an$ end{itemize} for all $i,j$, where $P=(p_{ij})$, $Q=(q_{ij})$, $V=(v_{ij})$, $W=(w_{ij})$ are bounded matrices on $l^2(Bbb Z)$ and $a,b,c,dinBbb C$. par It is clear that the solutions of the above system of equations introduces a class of infinite matrices whose entries are related ``dyadically". In cite{Ho:g}, it is shown that the seemingly complicated task of constructing these matrices can be carried out alternatively in a systematical and relatively simple way by applying the theory of Hardy classes of operators through certain operator equation on ${cal B}({cal H})$ (space of bounded operators on $cal H$) induced by a shift. Our purpose here is to present explicit formula for the homogeneous solutions this equation.