scattering of surface waves by a half immersed circular cylinder in fluid of finite depth release_2jyktulg6bfs5i2kgztjysa65i

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 "<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="$h$" id="E1"><mml:mi>h</mml:mi></mml:math>" 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<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="${a \mathord{\left/ {\vphantom {a h}} \right. \kern-\nulldelimiterspace} h}$" id="E2"><mml:mrow><mml:mi>a</mml:mi><mml:mo>/</mml:mo><mml:mi>h</mml:mi></mml:mrow></mml:math>starting with<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="$\left( {{a \mathord{\left/ {\vphantom {a h}} \right. \kern-\nulldelimiterspace} h}} \right)^2 $" id="E3"><mml:msup><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mrow><mml:mi>a</mml:mi><mml:mo>/</mml:mo><mml:mi>h</mml:mi></mml:mrow></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mn>2</mml:mn></mml:msup></mml:math>where "<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="$a$" id="E4"><mml:mi>a</mml:mi></mml:math>" is the radius of the circular cylinder. It is also shown that the coefficients of<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="$\left( {{a \mathord{\left/ {\vphantom {a h}} \right. \kern-\nulldelimiterspace} h}} \right)^2 $" id="E5"><mml:msup><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mrow><mml:mi>a</mml:mi><mml:mo>/</mml:mo><mml:mi>h</mml:mi></mml:mrow></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mn>2</mml:mn></mml:msup></mml:math>in these corrections do not vanish identically.
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