Let us consider the interior Robin problem for the Helmholtz equation in a starlike annulus
, whose boundaries are described by the polar equations respectively:
(12)
where denotes the propagation constant, are the outward-pointing normal unit vectors to the domain boundaries , respectively, and , are given regular weighting coefficients.
Under the mentioned assumptions, one can prove the following theorem.
Theorem
Let
(13)
and
(14)
where
(15)
being the usual Neumann symbol. Then boundary value problem (12) for the Helmholtz equation admits a classical solution such that the following Fourier-Hankel series expansion holds true:
(16)
For each index m, define
(17)
with denoting the Hankel function of kind and order m. Hence, the coefficients , in (16) can be determined by solving the infinite linear system
(18)
where
(19)
(20)
with and .
Proof Upon noting that in the stretched coordinate system ϱ, ϑ introduced in the x, y plane, the considered domain
turns into the circular annulus of radii a and b, one can readily adopt the usual eigenfunction method [10] in combination with the separation of variables (with respect to r and ϑ). As a consequence, elementary solutions of the problem can be searched in the form
(21)
Substituting into the Helmholtz equation, one easily finds that the functions , must satisfy the ordinary differential equations
(22)
(23)
respectively. The parameter μ is a separation constant whose choice is governed by the physical requirement that at any fixed point in the real plane the scalar field must be single-valued. So, by setting , one can easily find
(24)
where denote arbitrary constants. The radial function satisfying (23) can be readily expressed as follows:
(25)
with . Therefore, the general solution of Robin problem (12) can be searched in the form
(26)
Enforcing the Robin boundary condition yields
(27)
where
(28)
and
(29)
Hence, combining equations above and using the classical Fourier projection method, equations (17)-(20) follow after some algebraic manipulations. □
It is worth noting that the derived expressions still hold under the assumption that are piecewise continuous functions, and the boundary values are described by square integrable, not necessarily continuous, functions, so that the relevant Fourier coefficients , in equation (14) are finite quantities.