Boundary Value Problems for the Laplace Tidal Wave Equation
Abstract
This paper discusses the eigenvalue problem associated with the Laplace tidal wave equation (LTWE) given, for μ in (1,1), by (1μ ^{2}/μ ^{2τ ^{2}}y'(μ))' + 1/μ ^{2τ ^{2}}(s/τ[μ ^{2}+τ ^{2}/μ ^{2τ ^{2}}]+s^{2}/1μ ^{2}) y(μ ) = λ y(μ ), (LTWE) where s and τ are parameters, with s an integer and 0 < τ < 1, and λ determines the eigenvalues. This ordinary differential equation is derived from a linear system of partial differential equations, which system serves as a mathematical model for the wave motion of a thin layer of fluid on a massive, rotating gravitational sphere. The problems raised by this differential equation are significant, for both the analytic and numerical studies of SturmLiouville equations, in respect of the interior singularities, at the points ± τ , and of the changes in sign of the leading coefficient (1μ ^{2})/(μ ^{2}τ ^{2}) over the interval (1, 1) Direct sum space methods, quasiderivatives and transformation theory are used to determine three physically significant, wellposed boundary value problems from the SturmLiouville eigenvalue problem (LTWE), which has singular endpoints ± 1 and, additionally, interior singularities at ± τ . Selfadjoint differential operators in appropriate Hilbert function spaces are constructed to represent each of the three wellposed boundary value problems derived from LTWE and it is shown that these three operators are unitarily equivalent. The qualitative nature of the common spectrum is discussed and finite energy properties of functions in the domains of the associated differential operators are studied. This work continues the studies of LTWE made by earlier workers, in particular Hough, Lamb, LonguetHiggins and Lindzen.
 Publication:

Proceedings of the Royal Society of London Series A
 Pub Date:
 March 1990
 DOI:
 10.1098/rspa.1990.0029
 Bibcode:
 1990RSPSA.428..157H