Abstract
In this thesis a number of boundary value problems regarding the diffraction and propagation of long waves created in a water sheet in the presence of some combinations of vertical plane boundaries is studied. The whole system has been assumed to be on a rotating earth, except in the last problem (Chapter VI) where the rotation has been neglected. Also the differential equations governing the considered effect have been reduced to a linearised form. As regarding the history of the subject we mention that the dynamical theory of long waves progressing along a channel, when the geostrophic effect is ignored, was given by Lagrange in 1781. Lord Rayleigh in 1876 used the method of studying waves of constant form by considering the motion relative to the axes moving with the wave. The geostrophic effect in the study of the dynamics of small seas was introduced by Lord Kelvin (1879), who thus accounted for the transverse surface elevation across a channel, but his investigation was restricted to the so called harmonic Kelvin wave. In 1910 H. Poincare investigated some more general harmonic waves and his results were applied by H. Sverdrup to the tides of the Artic Sea in 1926. During and since World War II a great number of physical problems concerning surface waves in water have been investigated and various mathematical methods of solutions have been employed in this field of mathematical physics. Among the mathematical tools used, especially in the linearised case, are integral equation methods, conformal mapping and complex variable theory in general, the Laplace and Fourier transform techniques and methods employing Green's functions . In the present work the main methods used are based on the Wiener-Hopf technique. In all cases the problem is transferred directly to the Fourier transform domain where the boundary and continuity conditions are applied. This leads to a single functional equation for two unknown functions, the Wiener-Hopf equation, in the complex a-plane (a being the complex Fourier transform variable), which is solved by analytic continuation. This approach is suggested by Jones (1952) and has the advantage of bypassing the integral equation formulation also available, but more complicated in general, for solving problems of the type being examined (e. g. Copson, 1946). In Chapter I the basic differential equations of the motion of a perfect fluid on a rotating earth are derived and they are reduced to the linearised case of long waves. Chapter II deals with the possible modes which may exist in a rotating rectangular channel. The method applied is that of separation of variables. In Chapter III the problem of radiation of long waves from a rotating Channel with one semi-infinite and one infinite vertical barrier is investigated. An exact solution is obtained for the field inside the semi-infinite channel while the far field in the open region is evaluated asymptotically. The transmission and reflexion of long waves in a rotating bifurcated channel is developed in Chapter IV and an exact solution is derived for the whole field. Chapter V is concerned with the radiation and transmission of long waves from a rotating semi-infinite channel enclosed in parallel walls. Again an exact solution is found in an explicit form over the entire field. In the problems of Chapters III, IV and V the derived Wiener-Hopf equations are of the standard form. A generalised Wiener-Hopf equation arises in the problem of diffraction of a plane harmonic wave by a step which is discussed in Chapter VI. This equation can be solved only approximately because of the appearance of an infinite set of simultaneous linear algebraic equations. The field solution is given in an asymptotic form similar to that of the open region in the problem of Chapter III. The theory of the problem in Chapter III corresponds to the tidal wave leaving the North Sea as well as to the tide coming out of the Gulf of California. The models in Chapters IV and V are of less oceanographic interest but may have applications to the diffraction of tides in the East and West Indies. Finally, the results of the problem in Chapter VI may describe approximately the situation regarding the tidal waves which appear off the west coast of Africa. All the problems considered in this work are reduced to their analytical equivalents in acoustics and electromagnetism when we take the angular velocity of the rotating system equal to zero. Let it be noted lastly, that we specify in Appendix A a suitable branch of the function gamma = (alpha[2] - kappa[2])[1/2] arising from the transformation of the reduced wave equation (Helmholtz's equation) which governs the considered effect.