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III. WAVE FORCES The study of wave forces on coastal structures can be classified in two ways: (a) by the type of structure on which the forces act and (b) by the type of wave action against the structure. Fixed coastal structures can generally be classified as one of three types: (a) pile-supported structures such as piers and offshore platforms; (b) wall-type structures such as seawalls, bulkheads, revetments, and some breakwaters; and (c) rubble structures such as many groins, revetments, jetties and breakwaters. Individual structures are often combinations of these three types. The types of waves that can act on these structures are nonbreaking, breaking, or broken waves. Figure 7-66 illustrates the subdivision of wave force problems by structure type and by type of wave action and indicates nine types of force determination problems encountered in design. Figure 7-66. Classification of wave force problems by type of wave action and by structure type. Rubble structure design does not require differentiation among all three types of wave action; problem types shown as 1R, 2R, and 3R on the figure need consider only nonbreaking and breaking wave design. Horizontal forces on pile-supported structures resulting from broken waves in the surf zone are usually negligible and are not considered. Determination of breaking and nonbreaking wave forces on piles is presented in Section 1 below, Forces on Piles. Nonbreaking, breaking, and broken wave forces on vertical (or nearly vertical) walls are considered in Sections 2, Nonbreaking Wave Forces on Walls, 3, Breaking Wave Forces on Vertical Walls, and 4, Broken Waves. Design of rubble structures is considered in Section 7, Stability of Rubble Structures. NOTE: A careful distinction must be made between the English system use of pounds for weight, meaning force, versus the System International (SI) use of newtons for force. Also, many things measured by their weight (pounds, tons, etc.) in the English system are commonly measured by their mass (kilogram, metric ton, etc.) in countries using the SI system. 7-100 1. Forces on Piles . a. Introduction . Frequent use of pile-supported coastal and offshore structures makes the interaction of waves and piles of significant practical importance. The basic problem is to predict forces on a pile due to the wave-associated flow field. Because wave-induced flows are complex, even in the absence of structures, solution of the complex problem of wave forces on piles relies on empirical coefficients to augment theoretical formulations of the problem. Variables important in determining forces on circular piles subjected to wave action are shown in Figure 7-67. Variables describing nonbreaking, monochromatic waves are the wave height H , water depth d , and either wave period T , or wavelength L . Water particle velocities and accelerations in wave-induced flows directly cause the forces. For vertical piles, the horizontal fluid velocity u and acceleration du/dt and their variation with distance below the free surface are important. The pile diameter D and a dimension describing pile roughness elements e are important variables describing the pile. In this discussion, the effect of the pile on the wave- induced flow is assumed negligible. Intuitively, this assumption implies that the pile diameter D must be small with respect to the wavelength L . Significant fluid properties include the fluid density r and the kinematic viscosity n . In dimensionless terms, the important variables can be expressed as follows: dimensionless wave steepness dimensionless water depth ratio of pile diameter to wavelength (assumed small) relative pile roughness and a form of the Reynolds' number Given the orientation of a pile in the flow field, the total wave force acting on the pile can be expressed as a function of these variables. The variation of force with distance along the pile depends on the mechanism by which the forces arise; that is, how the water particle velocities and accelerations cause the forces. The following analysis relates the local force, acting on a section of pile element of length dz , to the local fluid velocity and acceleration that would exist at the center of the pile if the pile were not present. Two dimensionless force coefficients, an inertia or mass coefficient CM and a drag coefficient CD , are used to establish the wave-force relationships. These coefficients are determined by experimental 7-101 Figure 7-67. Definition sketch of wave forces on a vertical cylinder. measurements of force, velocity, and acceleration or by measurements of force and water surface profiles, with accelerations and velocities inferred by assuming an appropriate wave theory. The following discussion initially assumes that the force coefficients CM and CD are known and illustrates the calculation of forces on vertical cylindrical piles subjected to monochromatic waves. A discussion of the selection of CM and CD follows in Section e, Selection of Hydrodynamic Force Coefficients, CD and CM . Experimental data are available primarily for the interaction of nonbreaking waves and vertical cylindrical piles. Only general guidelines are given for the calculation of forces on noncircular piles. b. Vertical Cylindrical Piles and Nonbreaking Waves: (Basic Concepts) . By analogy to the mechanism by which fluid forces on bodies occur in unidirectional flows, Morison et al. (1950) suggested that the horizontal force per unit length of a vertical cylindrical pile may be expressed by the following (see Fig. 7-67 for definitions): (7-20) 7-102 where fi = inertial force per unit length of pile fD = drag force per unit length of pile r = density of fluid (1025 kilograms per cubic meter for sea water) D = diameter of pile u = horizontal water particle velocity at the axis of the pile (calculated as if the pile were not there) du/dt = total horizontal water particle acceleration at the axis of the pile, (calculated as if the pile were not there) CD = hydrodynamic force coefficient, the "drag" coefficient CM = hydrodynamic force coefficient, the "inertia" or "mass" coefficient The term fi is of the form obtained from an analysis of force on a body in an accelerated flow of an ideal nonviscous fluid. The term fD is the drag force exerted on a cylinder in a steady flow of a real viscous fluid is proportional to u2 and acts in the direction of the velocity u ; for flows that change direction this is expressed by writing u2 as u |u| ). Although these remarks support the soundness of the formulation of the problem as given by equation (7-20), it should be realized that expressing total force by the terms fi and fD is an assumption justified only if it leads to sufficiently accurate predictions of wave force. From the definitions of u and du/dt , given in equation (7-20) as the values of these quantities at the axis of the pile, it is seen that the influence of the pile on the flow field a short distance away from the pile has been neglected. Based on linear wave theory, MacCamy and Fuchs (1954) analyzed theoretically the problem of waves passing a circular cylinder. Their analysis assumes an ideal nonviscous fluid and leads, therefore, to a force having the form of fi . Their result, however, is valid for all ratios of pile diameter to wavelength, D/LA , and shows the force to be about proportional to the acceleration du/dt for small values of D/LA (LA is the Airy approximation of wavelength). Taking their result as indicative of how small the pile should be for equation (7-20) to apply, the restriction is obtained that (7-21) Figure 7-68 shows the relative wavelength LA /Lo and pressure factor K versus d/gT2 for the Airy wave theory. * * * * * * * * * * * * * * * * * * * EXAMPLE PROBLEM 16 * * * * * * * * * * * * * * * * * * GIVEN : A wave with a period of T = 5 s, and a pile with a diameter D = 0.3 m (1 ft) in 1.5 m (4.9 ft ) of water. 7-103 7-104 FIND : Can equation (7-20) be used to find the forces? SOLUTION : which, using Figure 7-68, gives LA = 0.47 Lo = 0.47 (39.0) = 18.3 m (60.0 ft) Since D/LA satisfies equation (7-21), force calculations may be based on equation (7-20). * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * The result of the example problem indicates that the restriction expressed by equation (7-21) will seldom be violated for pile force calculations. However, this restriction is important when calculating forces on dolphins, caissons, and similar large structures that may be considered special cases of piles. Two typical problems arise in the use of equation (7-20). (1) Given the water depth d , the wave height H , and period T , which wave theory should be used to predict the flow field? (2) For a particular wave condition, what are appropriate values of the coefficients CD and CM ? c. Calculation of Forces and Moments . It is assumed in this section that the coefficients CD and CM are known and are constants. (For the selection of CD and CM see Chapter 7, Section III,1,e, Selection of Hydrodynamic Force Coefficients CD and CM .) To use equation (7-20), assume that the velocity and acceleration fields associated with the design wave can be described by Airy wave theory. With the pile at x = 0 , as shown in Figure 7-67, the equations from Chapter 2 for surface elevation (eq. 2-10), horizontal velocity (eq. 2-13), and acceleration (eq. 2-15), are (7-22) 7-105 (7-23) (7-24) Introducing these expressions into equation (7-20) gives (7-25) (7-26) Equations (7-25) and (7-26) show that the two force components vary with elevation on the pile z and with time t . The inertia force fi is maximum for sin (- 2pt/T) = 1 , or for t = - T/4 for Airy wave theory.

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