The term hydromechanics normally refers to that part of fluid mechanics devoted to both the hydrostatics and hydrodynamics of incompressible flows. The term includes the effects of free surface at the air-sea interface. Although the focus of the discussion of hydrodynamic topics is on incompressible flows in this chapter, a discussion of hydrostatics includes the effects of the compressibility of seawater at great water depths. Because this is a review chapter, all aspects of hydrodynamics are not addressed. The reader is referred to the book by Robert A. Granger (1985) for an expanded coverage of the topics.

We begin our review of hydromechanics with a discussion of hydrostatics. Although this subject is basic to a course in fluid mechanics, hydrostatics is often neglected in favor of topics that are of more interest to the instructor. However, for the designer of deep-submergence vehicles, a thorough knowledge of the fundamentals of hydrostatics is required.

Hydrostatics

A discussion of hydrostatics must begin by paraphrasing Archimedes' Principle: A body placed in a liquid loses an amount of weight equal to the weight of the liquid that it displaces. From this simple observation, the hydrostatic equation can be derived. Consider the can buoy sketched in Figure 2.1, which displaces a volume (∨) of water. In that sketch, W is the buoy weight, A is the cross-sectional area, and d is the buoy draft.

Compliant Structures

Fixed ocean structures fall into three major types: rigid structures, compliant structures, and spread-footing structures. There are also two hybrid types of structures, called the tension-leg platform (TLP), which is a floating body held in place by high-tension mooring lines, and the articulated-leg platform (ALP), which is a quasi-rigid cylindrical hull attached to a universal joint on a sea-bed foundation. Rigid structures are usually designed for near-shore operations. Drilling structures used to tap the oil deposits in the shallow near-shore waters are normally of this type. As oil exploration moved further offshore, rigid structures became expensive and their cost-effectiveness decreased. To reduce drilling costs in the deeper water, compliant towers were constructed. One such structure is the Lena guyed tower, which was deployed in the Gulf of Mexico in waters of about 300 m in the 1980s. One advantage of the compliant structure is that its foundation is much less expensive compared to that of the rigid structure. The reason for this economy is that the wave loads are partially absorbed by the elasticity of the structure. In the North Sea, where severe storms occur over the entire year and the waters are relatively shallow, the spread-footing structure has been used to tap oil and gas deposits beneath the sea bed. These structures are monolithic structures, and simply rest on the bed. The structural load is distributed over the soil, reducing the normal load on the bed.

The majority of engineering problems encountered by ocean engineers involve wave-structure interactions. The waves in these interactions are altered and modified because of both the presence and motions of fixed and floating structures. The analyses of the wave-induced forces and motions of fixed structures require somewhat different mathematical tools than the analyses of wave-structure interactions of floating bodies. Wave-induced forces and motions of floating bodies are introduced in Chapter 10 and discussed in depth in Chapter 11.

The mathematical foundations on which contemporary analyses of wave-structure interactions are based date back to the nineteenth century when Stokes (1851) demonstrated that the total force on a body in an unsteady flow consisted of two components, those being a drag force and an inertial reaction of the fluid. The objects of the Stokes study were pendula (hanging circular cylinders and one with a spherical weight) moving in a viscous fluid where free-surface effects were not considered. Our interest begins with works done in the early part of the twentieth century, when most of the wave-structure interactions involved submerged horizontal circular cylinders. The orientations of the cylinders of interest were horizontal and fully submerged with their axes parallel to the wave fronts, as discussed by Havelock (1917), Lamb (1932), and others. In the Havelock and Lamb references, it is demonstrated how integral transforms and integral equations can be used to represent the free-surface response to a fully submerged, horizontal cylinder.

Ocean waves are caused by the motions of celestial bodies, seismic disturbances, moving bodies, and winds. The waves produced by these phenomena differ in size and character, and the consequences of each must be dealt with differently.

The gravitational attractions of both the moon and the sun cause the largest water waves, called the tides. The predictable tidal wave can be treated as a shallow-water wave because its length is much greater than the water depth. Extreme tides, called spring tides, occur when the attractive forces of both the moon and sun are aligned and in the same direction. These tides can cause flooding of lowlands if dikes or levees are not present. The tides can also be exploited by converting their energies into useable energy forms. This is normally accomplished by creating tidal barriers equipped with hydroturbines, taking advantage of the tidal-induced water level changes on opposite sides of the barriers. An excellent book on tidal energy conversion is that written by Charlier (1982).

Both sub-marine earthquakes and volcanic eruptions can produce a long, high-energy wave called a tsunami (the Japanese word for a tidal wave, although the wave referred to is not tidal in nature). This type of wave can pass a ship in the open ocean and not be noticed by the ship's crew because of the small wave height-to-wavelength ratio (called the wave steepness). As the tsunami approaches a land mass, the energy of the wave is transformed from mostly kinetic to mostly potential.

The study of nonlinear waves began early in the nineteenth century. Results of the first studies of these waves were used to determine the quasi-static wave loads on ships and other tethered floating structures. Although the first analytical techniques were rather simplistic, these techniques were used well into the twentieth century. In the mid-nineteenth century, more mathematically sophisticated analytical methods were introduced. These methods are used by both physical oceanographers and ocean engineers in their respective predictions of real wave properties and the time-dependent loadings on offshore structures.

There are a number of theories that can be used to approximately predict the properties of nonlinear waves. Probably the earliest of these theories is that of Gerstner (1808). The theory of Gerstner is a geometric type, and is commonly referred to as the trochoidal theory. This name results from the predicted profile of the breaking wave (see Figure 3.1e) which, from Gerstner's theory, is a trochoid, having a cusp at the crest. The trochoidal theory is rotational in a hydrodynamic sense (see Section 2.3), with the rotational direction being opposite of that actually observed. Results of the theory are still used today by some structural naval architects to predict the extreme quasi-static wave loads on ship hulls in both sagging (crests at both the bow and stern, and the trough amidships) and hogging (troughs at both the bow and stern, and a crest amidships).

It has been more than three decades since my first book on ocean engineering, Ocean Engineering Wave Mechanics. My purpose in writing that book was to give ocean engineering students and ocean technologists an introduction to the mechanics of water waves, and to present and demonstrate the analytical techniques used in wave-structure interaction problems. Since the 1973 publication of that book, ocean technology has been one of the most rapidly advancing engineering fields. The purpose of this book is to present both fundamental and advanced techniques in the analyses of both water waves and wave-structure interactions. The classical analytical works in the areas of wave mechanics are discussed in detail so that the reader can follow the lines of thought of the masters who produced these classic analyses.

Most of the material presented herein is for readers with a basic education in applied mechanics, including fluid mechanics or hydraulics and applied mathematics. The material is presented so that the reader can immediately apply the various analytical techniques to problems of interest. To this end, examples are presented in each section. Certain topics, such as the cnoidal theory, are of an advanced analytical nature and, as such, are more appropriate for postgraduate education. Following these topics are examples designed to demonstrate the application of these advanced analytical methods.

The purpose of this chapter is to give a cursory introduction to shore protection and to discuss some topics that are normally neglected in the coastal engineering literature. By shore protection, what is meant is the methodologies used in preventing either a net erosion or a net accretion of beach sand due to wave action. Specifically, some of the considerations that are part of the planning phase of shore protection projects are discussed. Shore protection is one of the topics under the broad heading of coastal engineering. The reader is referred to the books by Horikawa (1978), U.S. Army (1994), Goda (1975), Sorensen (1997), Dean and Dalrymple (2002), and that edited by Herbich (1999) for a more thorough discussion of the area of ocean engineering. In addition, the proceedings of the International Coastal Engineering Conference contain papers describing advances in both the science and technology applied to the coastal zone. These conferences occur about every two years, and are sponsored in part by the American Society of Civil Engineers (ASCE).

Shore Protection Methods

When a shoreline is identified as being unstable, the term usually means that there is a net loss of sand (erosion) or net gain (accretion). The engineer must determine whether or not the instability is short-term or long-term. Short-term instabilities are common, and are usually seasonal in nature. Winter waves tend to erode beaches, whereas summer waves tend to restore sand to the beaches.

In this chapter, basic analyses of the interactions of waves and compliant fixed and floating structures are presented. The reason for discussing both fixed and floating structures in a single chapter is that several of the analytical techniques are common to both. As in the previous chapters, the structural geometries studied here are those that can be dealt with on an analytical basis. The analyses of wave interactions with complicated structural geometries require the use of numerical techniques, such as finite-element analysis. Such situations are not addressed herein. The initial discussions of each type of structure are based on the assumption that the incident wave field is composed of regular, linear waves. These discussions are followed by considerations of structural motions in irregular (random), linear seas. Depending on water depth at the site, the wind fetch, and the wind duration, the irregular-wave analyses presented in Chapter 5 can be used for seas having a Beaufort Wind Force Scale up to 5. Sea states are used to quantify the severity of a sea, and are discussed in Chapter 1. The various sea-state scales are presented in Table 1.2.

Basic Concepts

As introduced in Section 9.2, the ambient water mass excited by an unsteadily moving body is called the added mass. The magnitude of the added mass is proportional to the inertial reaction force on the body resulting from the body motions. When the body moves close to the free surface, waves are also created.

The field of coastal engineering has many facets. The reader is referred to the handbooks edited by Herbich (1999) and by Kim (2009) for discussions of most of the coastal engineering areas. In this chapter the focus is on the coastal zone, where most of the attention of coastal engineers is focused on the effects of breaking and broken waves. The phenomenon of breaking is nonlinear in nature, as discussed in Section 4.6. The nonlinear behavior of breaking waves can be approximately predicted by theoretical analyses. The theoretical expressions for breaking waves presented in Section 4.6 are based on two assumptions. First, the water depth is assumed to be uniform, and second, the wave profile of the breaking wave is symmetric about a vertical plane containing the crest line. Even with these modeling constraints, the theoretical analyses have been found to have value in conceptual engineering design applications. Dean (1974) presents a detailed discussion of the limitations of the various wave theories when applied to waves at or near breaking.

After a shoaling wave breaks, energy losses occur, and the resulting behavior of the wave depends on phenomena that cannot be completely mathematically modeled. The behavior of the wave prior to the break is also affected by bed friction and, if the bed is porous, by percolation. These cause energy losses and complicate our ability to theoretically predict the behavior of the wave. Because of this, empirical formulas based on both experimental and field data have been developed.

The field of ocean engineering was formally identified as such in the 1960s. Prior to that decade, civil, mechanical, and electrical engineers and naval architects concentrated on ocean-related technologies in rather narrow areas. Two books that helped define ocean engineering in the 1960s are those by Wiegel (1964) and edited by Myers, Holm, and McAllister (1969). These books are still often referenced today. The integrated field of ocean engineering primarily resulted from the discovery of massive oil deposits beneath the sea beds. This discovery led to the increased production of both fixed and floating offshore structures and ancillary systems designed to support extraction systems for the energy resource.

Some of the contemporary ocean engineering areas are listed in Table 1.1. The areas discussed in this book are identified by an asterisk (*) in the table. The primary focus of this book is on wave-induced forces and the subsequent effects of ocean structures.

A large number of the engineering problems that must be faced in the design of ocean engineering systems involve water waves in one form or another. The engineer's ability to deal with these problems depends on the extent of their knowledge of the physics of water waves. In this book, basic and intermediate analytical techniques used in water-wave hydromechanics are presented. Each chapter contains a number of worked examples designed to help the reader better understand the various wave-related phenomena.

In Chapters 3 through 5, respectively, linear, nonlinear, and random waves are introduced, analyzed, and discussed. These waves are assumed to be affected by two physical boundaries, those being a flat, horizontal seafloor (or sea bed) and the free surface (at the air-water interface). In the present chapter, other boundaries are considered. These include vertical and sloping walls and sloping beds. The presence of these boundaries can cause the waves to be both modified (affecting the wave properties) and transformed (affecting the wave energy or energy flux). Specifically, the presence of boundaries causes waves to reflect, shoal, refract, and diffract. These wave phenomena and some of their engineering ramifications are discussed in the present chapter. An excellent “working document” covering the coastal engineering aspects of wave reflection, shoaling, refraction, and diffraction is the Shore Protection Manual of the Coastal Engineering Research Center (CERC) of the U.S. Army Corps of Engineers (see U.S. Army, 1984). A more recent CERC publication is the Automated Coastal Engineering System (User's Guide and Technical Reference), which is a computer-based document designed to assist coastal engineers in the predicting the behavior of waves (see Leenknecht, Szuwalski, and Sherlock, 1992). There are many other works available devoted to each wave phenomenon, the number being too large to individually reference in this chapter. For this reason, those works that are referred to in this chapter are those which are either encompassing or describe basic analyses, experiments, or prototype studies.