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The tidal vave function y(x) is a solution to an inhoraogeneous, linear, second-order differential equation with variable coefficient. Numerical values for the height-dependence terms, In the observed tides, have been utilized in finding y(x) as a solution to an initial-value problem. Complex Fast Fourier Transform technique is also used to obtain the solution in a complex form. Based on a realistic temperature structure, the atmosphere below 110 km has been divided into layers with distinct characteristics, and thus the technique of propagation in stratified media has been applied. The reduced homogeneous equation assumes the form of Helmholtz equation and with initial conditions the general solution is obtained, HIRAHARE TRIESTE August 1981 * To be submitted for publication. + To be presented at XV Biennial Fluid Dynamics Sympos-iura, to Be held in Poland 6-12 September 1981. ** Permanent address: Applied Math. Dept., Faculty of Science, Ain Shams Univ. Cairo, Egypt. In the development of the tidal theory(Siebert, 1961 and Chapman and Lindzen, 1970), the assumption is made such that the dependence of the tides on height, Co-latitude and longitude rf> are separable. The latitude-dependences of the tidal fields are expressed in terms of the Hough functions (H ) ( 6 ) , vhlcfi. are solutions of Laplace's tidal equation. It is an eigenvalue-eigenfunction problem and has been the subject of extensive study during the Ia3t decade CKato, 19-66; Lindzen, 1966; Longuett-Higgins, 1967 and Makarious, 1968). The vertical structure of the tidal wave propagation, on the other band, is expressed in terms of the wave function y(x), which is the solution of the Radial Wave equation. For a given mode of oscillation, the variable coefficient, in that equation, depends solely on the vertical temperature distribution. Previous investigators tPekerls, 1937; Jacchia and Kopal, 1952; Butler and Small, 1963 and Lindzen; 1967) adopted,however, simplified vertical temperature structure in order to render the mathematical treatment more tractable. Consequently classical tidal theory has limitations, arising out of the assumptions on which it is based, which need to be recognized when making close comparison with the observational data (Groves, 1977). With a realistic temperature structure, however, no closed form solution to the vertical wave equation exists and the solution has to be approached numerically. This is the goal of the present study. Analytical expressions for the general solution, in conformity with the observed tidal oscillations In the atmosphere, have been obtained. The numerical solutions for the oscillatory and trapped modes of tidal wave propagation have also been presented and discussed.

A Numerical Solution to The Radial Equation of the Tidal Wawe Propagation