T. L. Yip T. Sahoo A. T. Chwang Fellow ASME Department of Mechanical Engineering, The University of Hong Kong, Pokfulam Road, Hong Kong Wave Oscillation in a Circular Harbor With Porous Wall The wave resonance in a circular harbor surrounded by a porous seawall is analyzed. Matching the velocity and pressure along the porous seawall and the harbor entrance, the full solution is obtained. The resonance condition is found to depend on the wave fre- quency, the complex porous-effect parameter and the internal dimension of the porous seawall. The oscillation characteristics are analyzed in different cases. The condition for natural oscillation is derived by studying the wave resonance in a closed circular harbor surrounded by a porous seawall. DOI: 10.1115/1.1379955 1 Introduction In the last decade, there has been a significant change in the nature of harbor traffic in Hong Kong, which results in the dete- rioration of wave conditions in Victoria Harbor. As a result, the dynamic and mooring forces acting on ships and docks are seri- ously affected by the high wave oscillation and in turn create serious problems to different marine structures, affecting loading and unloading of cargoes. As a means to dissipate wave energy, a porous seawall is introduced inside an existing harbor, which will reduce the wave oscillation and improve the general wave climate. The vertical-wall harbors are widely used for their simple de- sign and construction. In the presence of waves, a vertical harbor wall reflects most of the wave energy incident on it. With strong harbor oscillations, vertical walls are subjected to large wave forces. Recently, permeable breakwaters, detached breakwaters, and submerged breakwaters have received much attention and their capability to dissipate wave energy is widely studied. Some of the energy-dissipating breakwaters are being tested in harbors 1,2. On the other hand, wave agitation in harbor due to an incoming wave of a particular frequency may last for a long time. This agitation leads to a resonant state and is the cause of extremely high wave oscillations inside the harbor. The dynamic and moor- ing forces acting on marine structures are increased during this high oscillation which usually create serious problems to loading and unloading of cargoes. Thus, during the harbor planning, mea- sures should be taken to avoid such harbor resonance. There are two kinds of oscillations existing in a harbor, one is the free os- cillation and the other forced oscillation. Lamb 3 analyzed the effect of free oscillation in closed rectangular, circular, and ellip- tical basins. McNown 4 investigated the forced oscillation in a circular harbor having a narrow opening. In a rectangular harbor, the effect of forced oscillation was analyzed by Kravtchnenko and McNown 5. Further study on harbor resonance was done by Miles and Munk 6, LeMehaute 7 and Ippen and Goda 8. Miles and Munk 6 found that the wider the harbor mouth, the smaller the amplitude of the resonant oscillation which is contra- dictory to the fact that less wave energy will be transmitted to the harbor through a smaller opening. This phenomenon was known as the harbor paradox. Lee 9 considered rectangular and circular harbors with their openings located on a straight coastline while Mei and Petroni 10 dealt with a circular harbor protruding half- way into the open sea. To deal with arbitrary harbor configuration, Hwang and Tuck 11 and Lee 9 developed integral equation methods while Mei and Chen 12 provided a hybrid element method. Numerical studies for harbors of arbitrary geometry have also been verified by field and experimental data e.g. 9,11. Recently, certain amount of numerical work is available to include the reflectivity of the harbor wall 13,14. However, because of the deficiency of numerical methods, the results can only repre- sent special conditions, not the general relationship between the reflectivity and the harbor oscillation. In recent times, porous breakwaters are being constructed for dissipating wave energy in order to reduce the hydrodynamic forces on breakwaters. With the assumption of Darcy’s law, Sollitt and Cross 15 and Chwang 16 separately developed models to study the flow past porous structures. These methods were unified and combined by Yu and Chwang 17 to become the most ac- ceptable one in the recent literature of flow past porous structures. Yu and Chwang 17 studied the problem of wave resonance in a harbor with a porous breakwater. It is observed that a porous breakwater can reduce the amplitude of resonant frequency sig- nificantly. A small but finite permeability of the breakwater is found to be optimal to diminish the resonant oscillation. In the present paper, we investigate the problem of wave reso- nance in a circular basin surrounded by a porous breakwater. The basin has an entrance located on a straight coastline. As a particu- lar case, the wave resonance in a closed circular basin surrounded by a permeable breakwater is analyzed. Matching the velocity as well as the pressure along the porous seawall and the harbor en- trance, the full solution is obtained and the resonance condition is derived. The effect of the porous-effect parameter and the position of the breakwater on wave oscillation are analyzed. The present work should be useful in future harbor design and modifications. 2 Formulation of the Problem The problem under consideration is three dimensional in nature and is studied in a cylindrical coordinate system with uniform depth h. The opening of the harbor is along the coastline at a distance b cos  from the center of the harbor with 2 being the opening angle of the harbor see Fig. 1. The circular harbor is of radius b with an inner permeable circular wall of radius a. Assum- ing that the fluid is inviscid and incompressible and its motion irrotational, we can define a velocity potential ( r ,  , z , t ) which satisfies the Laplace equation. Assuming the motion is simple har- monic in time, we can express  as ( r ,  , z , t ) =Re (r,,z)e -it  with  being the angular frequency. The fluid domain is divided into three regions: i the open sea region, ii the region between the porous wall and the solid harbor wall and iii the inner harbor region surrounded by the porous wall.  j ( j =1,2,3) denotes the velocity potential in region j. The spa- tial velocity potential  satisfies the Laplace equation Contributed by the Applied Mechanics Division of THE AMERICAN SOCIETY OF MECHANICAL ENGINEERS for publication in the ASME JOURNAL OF APPLIED ME- CHANICS. Manuscript received by the ASME Applied Mechanics Division, June 24, 2000; final revision, Sept. 26, 2000. Associate Editor: D. A. Siginer. Discussion on the paper should be addressed to the Editor, Prof. Lewis T. Wheeler, Department of Mechanical Engineering, University of Houston, Houston, TX 77204-4792, and will be accepted until four months after final publication of the paper itself in the ASME JOURNAL OF APPLIED MECHANICS. Copyright © 2001 by ASME Journal of Applied Mechanics JULY 2001, Vol. 68 Õ 603 Downloaded 28 Dec 2009 to 140.121.146.141. Redistribution subject to ASME license or copyright; see http://www.asme.org/terms/Terms_Use.cfm