Abstract
A train of surface waves is normally incident on a half immersed circular cylinder in a fluid of finite depth. Assuming the linearized theory of fluid under gravity an integral equation for the scattered velocity potential on the half immersed surface of the cylinder is obtained. It has not been found possible to solve this in closed form even for infinite depth of fluid. Our purpose is to obtain the asymptotic effect of finite depth “h” on the transmission and reflection coefficients when the depth is large. It is shown that the corrections to be added to the infinite depth results of these coefficients can be expressed as algebraic series in powers of a/h starting with (a/h) 2 where “a” is the radius of the circular cylinder. It is also shown that the coefficients of (a/h) 2 in these corrections do not vanish identically.
Highlights
It has not been found possible to solve this in closed form even for infinite depth of fluid
Our purpose is to obtain the asymptotic effect of finite depth "h" on the transmission and reflection coefficients when the depth is large
It is shown that the corrections to be added to the infinite depth results of these coefficients can be expressed as algebraic series in powers of a/h starting with (a/h) 2 where "a" is the radius of the circular cylinder
Summary
This paper deals with the reflection and transmission of surface waves normally incident on an infinitely long cylinder with circular cross section of radius "a" and horizontal generators, half immersed in an ideal fluid of infinite horizontal extent but finite constant depth "h." The two-dimensional linearized theory of an ideal fluid under gravity is assumed to hold in the fluid region. Dean and Ursell [2] investigated the interaction of a fixed half innersed circular cylinder with a train of surface waves in a fluid of inflnite depth They expressed the potential in the fluid region as a sum of wave source petentials and multiple singularities at the center of the circle with unknown coefficients and obtained an infinite set of linear algebraic equations involving the unknown coefficients. Mei and Black [6] considered scattering of surface waves by rectangular obstacles fixed at the bottom or at the free surface in a fluid of finite depth and applied variational techniques to obtain the reflection and transmission coefficierts numerically. For large h, ko K (I + 2e-2 kh), so that k and K differ by an exponentially small quantity for large h
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