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This manual is in English. Fields marked are sourced values that have been evaluated by the manufacturer or a third party, all others are design specifications. David K. Language s List all languages. Reconstruction or development? Is the manual intended for a response effort or for longer term development.
Volume I Chapter 1. Introduction Chapter 2. Historical Perspective Chapter 3. Identifying Hazards Chapter 4. Siting Chapter 5. Investigating Regulatory Requirements Chapter 6. Pre-Design Considerations Chapter 8. Determining Site-Specific Loads Chapter 9.
Designing the Building Chapter For long period waves in relatively shallow water this boundary layer can extend up through much of the water column. But, for typical wind waves the boundary layer is quite thin relative to the water depth, and if propagation distances are not too long and the bottom is not too rough, bottom friction energy losses can be neglected. Bottom Percolation If the bottom is permeable to a suYcient depth, the wave-induced Xuctuating pressure distribution on the bottom will cause water to percolate in and out of the bottom and thus dissipate wave energy.
Bottom Movement When a wave train propagates over a bottom consisting of soft viscous material such as the mud deposited at the Mississippi River Delta the Xuctuating pressure on the bottom can set the bottom in motion. Viscous stresses in the soft bottom dissipate energy provided by the waves. Wave Group Celerity Consider a long constant-depth wave tank in which a small group of deep water waves is generated. As the waves in the front diminish in height, new waves will appear at the rear of the group and commence to grow.
One new wave will appear each wave period so the total number of waves in the group will continually increase. This phenomenon causes the wave group to have a celerity that is less than the celerity of the individual waves in the group. Since the total energy in the group is constant neglecting dissipation the average height of the waves in the group will continually decrease. An explanation for this phenomenon can be found in the fact that only a fraction [n; see Eq. Thus, the Wrst wave in the group is diminished in height by the square root of n during the advance of one wave length.
Waves in the group lose energy to the wave immediately behind and gain energy from the wave in front. The last wave in the group leaves energy behind so, relative to the group, a new wave appears each T seconds and gains additional energy as time passes. A practical consequence of the deep water group celerity being less than the phase celerity of individual waves is that when waves are generated by a storm, prediction of their arrival time at a point of interest must be based on the group celerity.
To develop an equation for calculating the group celerity Cg consider two trains of monochromatic waves having slightly diVerent periods and propagating in the same direction.
The superimposition of the two wave trains results in a beating eVect in which the waves are alter- nately in and out of phase. This produces the highest waves when the two components are in phase, with heights diminishing in the forward and backward directions to zero height where the waves are exactly out of phase.
The result is a group of waves advancing at a celerity Cg. If you follow an individual wave in the wave group its amplitude increases to a peak and then diminishes as it passes through the group and disappears at the front of the group. Two wave trains shown separately and superimposed. The group advances a distance dx in the time dt, where dx is the distance traveled by the group in the time interval dt minus the one wave length that the peak wave dropped back as the in-phase wave drops back one wave length each period.
For a general relationship for the group celerity, employing the dispersion relationship with Eq. Another way to look at this is that the wave energy is propagated forward at the group celerity.
Problems commonly addressed by the application of radiation stress include the lowering setdown and raising setup of the mean water level that is induced by waves as they propagate into the nearshore zone, the interaction of waves and currents, and the alongshore current in the surf zone induced by waves obliquely approaching the shore. Radiation Stress The horizontal Xux of momentum at a given location consists of the pressure force acting on a vertical plane normal to the Xow plus the transfer of momen- tum through that vertical plane.
The latter is the product of the momentum in the Xow and the Xow rate across the plane. From classical Xuid mechanics, the momentum Xux from one location to another will remain constant unless there is a force acting on the Xuid in the Xow direction to change the Xux of momentum. In Eq. The overbar denotes that the Wrst term on the right must be averaged over the wave period. Inserting the pressure and the particle velocity from Eq.
Note that in deep water Eqs. Wave Setup When a train of waves propagates toward the shore, at some point, depending on the wave characteristics and nearshore bottom slope, the waves will break. Landward of the point of wave breaking a surf zone will form where the waves dissipate their energy as they decay across the surf zone. As the waves approach the breaking point there will be a small progressive set down of the mean water level below the still water level.
This setdown is caused by an increase in the radiation stress owing to the decreasing water depth as the waves propagate toward the shore. The setdown is maximum just seaward of the breaking point.
In the surf zone, there is a decrease in radiation stress as wave energy is dissipated. This eVect is stronger than the radiation stress increase owing to continued decrease in the water depth. The result is a progressive increase or setup of the mean water level above the still water level in the direction of the shore. This surf zone setup typically is signiWcantly larger than the setdown that occurs seaward of the breaking point.
The equations that predict the wave-induced nearshore setdown and setup can be developed by considering the horizontal momentum balance for two-dimen- sional waves approaching the shore Longuet-Higgins and Stewart, Force balance for wave-induced setup analysis.
Consider Figure 2. The forces and related change in the radiation stress at the boundaries are as shown. This equation applies to the regions before and after the breaking point. For the region just seaward of the breaking point assume that the wave power is constant and employ Eq. This rate of energy dissipation is complex and typically nonuniform. However, to reasonably develop an equation for wave setup, we will assume that the wave height across the surf zone is proportional to the depth below the local mean water level, i.
A reasonable value for g is 0. These assumptions lead to a solution to Eq. The nearshore bottom slope through the surf zone is 0. Find the setdown at the breaker point and the setup above the still water line at the still water line contour of the shore.
Assume shallow water wave conditions throughout. However, higher incident waves will break further seaward so the same mean water level slope will yield a higher mean water level throughout the surf zone. However, experiments conducted by Saville in a large two- dimensional wave tank yielded results that favorably agree with predictions from these equations.
Also, the equations apply to waves approaching normal to the shore. If the waves approach obliquely to the shore, only the shore normal component of radiation stress will induce setdown and setup. When the reXected wave passes through the incident wave a standing wave will develop. It is worthwhile to investigate the nature of wave reXection and standing waves, particularly the resulting surface proWle and particle kinematics of the resulting wave motion as well as the dependence of the reXected wave charac- teristics on the reXecting structure makeup.
When these two waves are superim- posed the resulting motion is a standing wave as depicted in Figure 2. The water surface oscillates from one position to the other and back to the original position in one wave period.
The arrows indicate the paths of water particle oscillation. Under a nodal point particles oscillate in a horizontal plane while under an antinodal point they oscillate in a vertical plane. When the surface is at one of the two envelope positions shown, water particles instantaneously come to rest and all of the wave energy is potential.
Halfway between the envelope positions the water surface is horizontal and all wave energy in kinetic.
The net energy Xux if the two component waves are identical is zero. The velocity potential for a standing wave can be obtained by adding the velocity potentials for the two component waves that move in opposite direc- tions. Standing wave particle motion and surface proWle envelope. If the component progressive wave heights are H, the standing wave height is 2H. The pressure is hydrostatic under a node where particle acceleration is horizontal; but under an antinode there is a Xuctuating vertical component of dynamic pressure.
Wave ReXection In a standing wave, the particle velocity under an antinode is always vertical. If a frictionless, rigid, vertical, impermeable wall were placed at the antinode the water particle motion would be unaVected. Thus, we would have a standing wave caused by the reXection of a progressive wave from the wall. The particle velocity and the pressure distribution along the wall would be given by Eqs. The surface proWle and the particle motion in this standing wave are depicted in Figure 2.
Considering Figure 2. It can also be shown the reXec- tion coeYcient equals the diVerence between the two envelope heights divided by the sum of the two envelope heights. When wave tank tests are being run with monochromatic waves and a reXecting structure, the wet mark on the side of the tank displays the upper envelope shown in Figure 2.
A wave gage mounted on a carriage and slowly moved at least one wave length along the wave tank will measure the node and antinode envelope heights which can be used to calculate the reXection coeYcient for a monochromatic wave. Further on an asymmetry also develops around a vertical axis through the wave crest neither asymmetry is deWned by the small amplitude wave theory. These asymmetries ultimately lead to wave instability and breaking. ProWle Asymmetry Figure 2.
Besides the vertical asymmetry resulting in a crest amplitude that exceeds half the wave height, the front face of the wave becomes steeper than the back face and the distance in the direction of wave propagation from crest to trough is less than the distance from trough to crest.
These asymmetries increase as the wave moves into shal- lower and shallower water. They also contribute to increased particle velocities at the wave crest and ultimately to crest instability and wave breaking. DeWnition of proWle asymmetry terms. Wave tank experiments were conducted by Adeyemo for intermediate depth waves shoaling on slopes from to These slopes are somewhat steeper than found in most nearshore areas. He presented his data in terms of four values deWned as follows see Figure 2.
The experiments showed the vertical asymmetry continuously increased as the wave shoaled, reaching a maximum of between 0.
Flatter slopes mean that the wave has more travel time for the asymmetry to develop. Thus, for natural beach slopes that are Xatter than the experimental slopes one might expect vertical asymmetries greater than the 0.
The slope and horizontal asymmetries also continuously increased as the wave shoaled; but, as opposed to vertical asymmetries steeper bottom slopes caused greater slope and horizontal asymmetries. Wave Breaking If a wave has suYcient height in any water depth it will break.
In deep water, for a given wave period, the crest particle velocity is proportional to the wave height. From the small-amplitude wave theory, the wave celerity is independent of the wave height. So, as the wave height increases the crest particle velocity will eventually equal the wave celerity and the wave will break.
In shallow water, as the water depth decreases the crest particle velocity increases and the wave celerity decreases, leading to instability and breaking. As a consequence, Eq.
For deep water Eq. No matter how high the deep water wind generated waves are, the highest wave that can reach the structure is dependent primarily on the water depth in front of the structure. Thus, as structures are extended further seaward they tend to be exposed to higher, more damaging waves. Waves breaking on a beach are commonly classiWed into three categories U. These three breaker classes are: Spilling. As breaking commences, turbulence and foam appear at the wave crest and then spread down the front face of the wave as it propagates toward the shore.
The turbulence is steadily dissipating energy, resulting in a relatively uniform decrease in wave height as the wave propagates forward across the surf zone. The wave crest develops a tongue that curls forward over the front face and plunges at the base of the wave face. Wave breaker classiWcation. The plunging tongue of water may regenerate lower more irregular waves that propagate forward and break close to the shore.
The crest and front face of the wave approximately keep their asymmetric shape as they surge across the beach slope. This form of breaking is a progression toward a standing or reXecting wave form. While the above three classes are relatively distinct, for gradually changing incident wave steepnesses and bottom slopes there is a gradual transition from one form to the next. Some investigators add a transitional class—collapsing breakers—between plunging and surging.
Only spilling and plunging breakers occur in deep water and they are the most common types of breakers in shallow water.
The type of breaker is important, for example, to the stability of a stone mound structure exposed to breaking waves.
As discussed above, Eq. A number of experimenters have investigated nearshore breaking conditions in the laboratory and presented procedures for predicting the breaking height Hb and water depth at breaking db as a function of incident wave characteristics and bottom slope m. Figures 2. Army Coastal Engineering Research Center and based on studies by Goda and Weggel , are commonly used for estimating breaking conditions.
Given the beach slope, the unrefracted deep water wave height, and the wave period one can calculate the deep water wave steepness and then determine the breaker height from Figure 2. The regions for the three classes of wave breaker types are also denoted on this Wgure. With the breaker height one can then determine the water depth at breaking from Figure 2. If a wave refracts as it propagates toward the shore, the equivalent unrefracted wave height given by 3.
Dimensionless breaker height and class versus bottom slope and deep water steepness. ModiWed from U. Army Coastal Engineering Research Center, Dimensionless breaker depth versus bottom slope and breaker steepness. Douglass conducted limited laboratory tests on the eVect of inline fol- lowing and opposing winds on nearshore wave breaking.
He found that oVshore directed winds retarded the growth of wave height toward the shore and conse- quently caused the waves to break in shallower water than for the no wind condition. Onshore winds had the opposite eVect but to a lesser extent. For the same incident waves, oVshore winds caused plunging breakers when onshore winds caused waves to spill.
The design of some coastal structures is dependent on the higher wave that breaks somewhat seaward of the structure and plunges forward to hit the structure. Thus, when designing a structure for breaking wave conditions, the critical breaking depth is some point seaward of the structure that is related to the breaker plunge distance Xp as depicted in Figure 2. DeWnition sketch for breaker plunge distance. DeWnition sketch for wave runup.
Prediction of the wave runup is important, for example, for the determination of the required crest elevation for a sloping coastal structure or to establish a beach setback line for limiting coastal construction. The runup depends on the incident deep water wave height and period, the surface slope and proWle form if not planar, the depth ds fronting the slope see Figure 2.
See U. Army Coastal Engin- eering Research Center, for similar plots for other slope conditions. Figure 0 2. Also, for most beach and revetment slopes which are Xatter than 1 on 2 , the wave runup increases as the slope becomes steeper. Table 2. The factor r is the ratio of the runup on the given surface to that on a smooth impermeable surface and may be multiplied by the runup determined from Wgures such as Figure 2. The wave breaks and runs up on a grass covered slope having a toe depth of 4 m.
Determine the breaking wave height and the wave runup elevation on the grass-covered slope. From Figure 2. A summary follows: 1. For most typical bottom slopes the dispersion equation is satisfactory for predicting the wave celerity and length up to the breaker zone.
For increasing beach slopes and wave steepnesses, the wave height predic- tions given by Eq. This discrepancy increases as the relative depth decreases. As an example, on a slope, for a relative depth of 0. For waves on a relatively Xat slope and having a relative depth greater than about 0. Limitations of the small-amplitude theory in shallow water and for high waves in deep water suggest a need to consider nonlinear or Wnite-amplitude wave theories for some engineering applications.
The next chapter presents an over- view of selected aspects of the more useful Wnite-amplitude wave theories, as well as their application and the improved understanding of wave characteristics that they provide. Airy, G. Battjes, J. Webber and G. Douglass, S. Eagleson, P. Ippen, A. LeMehaute, B. Longuet-Higgins, M. Miche, M.
Saville, T. Weggel, J. A two-dimensional wave tank has a still water depth of 1. The tank is 1 m wide. A wave generator produces monochromatic waves that, when measured at a wave gage installed before the toe of the slope, have a height of 0. If not, what would the equivalent deep water length, celerity, group celerity, energy, and energy density be? Compare these values to those in part a. An ocean bottom-mounted pressure sensor measures a reversing pressure as a train of swells propagates past the sensor toward the shore.
The pressure Xuctuations have a 5. Calculate and plot the wave height as a function of position from deep water into the point at which the wave breaks.
OVshore, in deep water, a wave gage measures the height and period of a train of waves to be 2 m and 7. The wave train propagates toward the shore in a normal direction without refracting and the nearshore bottom slope is It passes the outer end of a pier located in water 4. Is this a deep, transitional or shallow water wave at the end of the pier? What will the wave height be as the wave breaks? What type of breaker will it be? This wave is one of a train of waves that is traveling normal to the shore without refracting.
The bottom slope is A wave has a height of 1. Plot the horizontal component of velocity, the vertical component of acceler- ation, and the dynamic pressure at a point 2 m below the still water level versus time for a 6-s interval. Plot the three values on the same diagram and comment on the results. Estimate the maximum height wave that can be generated in a wave tank having water 1. A wave has a measured height of 1. If it shoals on a slope how wide will the surf zone be?
Assume the wave propagates normal to the shore without refracting. Determine the height and period of the wave causing the measured pressure Xuctuation. Derive an equation for the horizontal component of particle convective acceleration in a wave. Compare the horizontal components of convective and local acceleration versus time for a time interval of one wave period, at a distance of 2 m below the still water level and for a 1 m high 6 s wave in water 5 m deep.
Demonstrate, using Eq. Derive the equations for the horizontal and vertical components of particle acceleration in a standing wave, starting from the velocity potential [Eq. As the tide enters a river and propagates upstream, the water depth at a given location is 3. At this location the tide range is 1 m. If the dominant tidal component has a period of Consider the conditions given in Problem At a location in the river where the water depth is 5.
The Wrst wave of a group of waves advancing into still water is 0. The water depth is 4. How high is this wave 8 s later? Consider a 1 m high, 4 s wave in water 5 m deep.
Plot suYcient velocity potential lines to deWne their pattern and then sketch in orthogonal streamlines. Consider a deep water wave having a height of 2. Calculate the wave celerity just prior to breaking and compare it to the crest particle velocity.
Comment on the reason for any discrepancies. A 12 s, 2 m high wave in deep water shoals without refracting. Calculate the maximum horizontal velocity component and the maximum horizontal displacement from the mean position for a particle 5 m below the still water level in deep water and where the water depth is 6 m.
As the wave given in Problem 4 propagates toward the shore determine the mean water level setdown at the breaker line and the setup 40 m landward of the breaker line. From this, derive the potential energy per wave length and the potential energy density, both as a function of time. Realizing that at the instant a standing wave has zero particle velocity throughout, all energy is potential energy, determine the total and kinetic energies per wave length and the total and kinetic energy densities.
For the conditions in Example 2. A wave having a height of 2. A water particle velocity of 0. At what water depth will sand movement commence as the wave shoals? Using shallow water wave equations for celerity and water particle vel- ocity and the criteria that at incipient breaking the crest particle velocity equals the wave celerity, derive a criterion for wave breaking.
Comment on the result of this derivation. To an observer moving in the direction of a monochromatic wave train at the wave celerity, the wave motion appears to be steady.
Apply the Bernoulli equation between these two points to derive Eq. For a given wave height and period and water depth which of the follow- ing wave parameters depend on the water density: celerity, length, energy dens- ity, particle pressure, and particle velocity at a given depth? Explain each answer. How does increased water viscosity aVect a wave train as it propagates toward the shore? Waves with a period of 10 s and a deep water height of 1 m arrive normal to the shore without refracting.
A m long device that converts wave motion to electric power is installed parallel to the shore in water 6 m deep. Demonstrate that the velocity potential deWned by Eq. A wave with a period of 7 s propagates toward the shore from deep water.
Using the limit presented in this chapter, at what water depth does it become a shallow water wave? If the deep water wave height is 1 m would this wave break before reaching the shallow water depth? Assume that no refraction occurs and that the nearshore slope is But the two surface boundary conditions had to be linearized and then applied at the still water level rather than at the water surface. There is no general solution to the Laplace equation and three gravity wave boundary conditions.
All wave theories require some form of approximation or another. The intent of this chapter is to provide a brief overview of selected useful Wnite- amplitude wave theories and to employ these theories to provide additional insight into the characteristics of two-dimensional waves.
A discussion of the application of these Wnite-amplitude theories to engineering analysis will also be presented. There are numerical theories that employ a Wnite diVerence, Wnite element, or boundary integral method to solve the Laplace and boundary condition equations. There are also analytical theories in which the velocity potential and other parameters such as the surface amplitude and wave celerity is written as a power series that is solved by successive approximations or by the perturbation approach.
On the other hand, the analytical theories produce speciWc equations for the various wave characteristics which are given in terms of the wave height and period and the water depth. Both numerical and analytical theories are not complete solutions of the wave boundary value problem, but inWnite series solutions that must be truncated at some point e.
In this chapter we brieXy consider four Wnite-amplitude wave theories. The Stokes theory for deep water waves and the cnoidal and solitary theories for shallow water waves are useful analytical theories. For a more detailed presentation of these and other Wnite-amplitude wave theories see Wiegel , Ippen , Sarpkaya and Isaacson , and Dean and Dalrymple If any of the Wnite-amplitude wave theories are to be used in practice, two important considerations arise.
The Wrst is the choice of which theory to use to calculate wave characteristics for a given combination of wave height and period and water depth. As indicated above, most Wnite-amplitude theories are devel- oped for a speciWc range of relative depths and the higher order solutions for a particular theory are often more appropriate for higher steepness waves.
In some cases, one characteristic may be best predicted by one theory and another by a diVerent theory. The Wnite-amplitude theories are generally much more complex and diYcult to apply than the small-amplitude theory, but generally yield better results.
Is the increased eVort justiWed given the accuracy of analysis desired and the accuracy to which the input wave conditions are known? A second consideration is the diYculty of employing Wnite-amplitude wave theories to calculate wave transformations over a wide range of water depths, since these theories are commonly developed for speciWc ranges of relative water depth.
In practice, wave conditions are typically forecast for an oVshore deep water location and then must be transformed to some nearshore point for coastal design analysis. This factor, plus the ease in applying the theory, generally induces the coastal engineer to apply the small-amplitude wave theory for most analyses. These two considerations are addressed further in Section 3. As the deep water wave steepness increases improved accuracy can generally be achieved at the price of more onerous equations to work with if the Stokes theory is carried out to higher orders.
Various higher order approximations to the Stokes theory have been developed. For example, see Skjelbreia for a third-order theory, Skjelbreia and Hendrickson for a Wfth-order theory, and Schwartz for much higher order solutions based on calculations using a powerful computer.
For engineering applications the second- order and possibly the Wfth-order theories are most commonly used. Comparison of Eq. The magnitude of the second term on the right is dependent on the wave steepness, a ratio that has a numerical value that is signiWcantly less than unity but that increases as the wave amplitude increases for a given wave period. The second term on the right also has a frequency that is twice that of the small-amplitude term.
The eVect of the second-order term having twice the frequency of the small-amplitude or Wrst-order term is that the two components of surface amp- litude reinforce i. This yields a surface proWle vertical asymmetry more peaked wave crest and Xatter wave trough than a cosine proWle given by the small-amplitude theory that grows as the wave steepness increases. Thus, to the second order, waves are still period dispersive but not amplitude dispersive.
For the same wave period higher waves travel faster than lower waves. For deep water, Eq. Equation 3. Example 3. Using the equations presented above, calculate the wave celerity and length. Also, determine the wave crest and trough amplitudes. Compare the results to those from the small-amplitude wave theory. The Stokes theory values are both 5. From Eq. A point is reached where the trough surface becomes horizontal.
Increases in wave steepness beyond this point cause a hump to form and grow at the wave trough. This hump is not a real wave phenomenon and its appearance is an indication that the theory is being used beyond its appropriate limit. This puts a signiWcant restriction on the use of the second-order theory as the wave propagates into shallower water. The particle velocity and acceleration are increased under the wave crest and diminished under the wave trough.
Again, these asymmetries increase as the wave steepness increases. This mass transport is also evident in the second-order particle displacement equations. If we divide the last term in Eq.
Compare this to the wave celerity and crest particle velocity. Solution: For the Wrst or second order, the wave length is given by Eq.
Then, the mass transport velocity, given by Eq. Away from the bottom there is a time average vertical momentum Xux owing to the crest to trough asymmetry in the vertical velocity component.
This produces the above-zero time average dynamic pressure given by this last term on the right in Eq. Keulegan recommended a range for Stokes theory application extending from deep water to the point where the relative depth is approximately 0.
However, the actual Stokes theory cutoV point in intermediate water depths depends on the wave steepness as well as the relative depth. For steeper waves, the higher order terms in the Stokes theory begin to unrealistically distort results at deeper relative depths.
For shallower water, a Wnite-amplitude theory that is based on the relative depth is required. Cnoidal wave theory and in very shallow water, solitary wave theory, are the analytical theories most commonly used for shallower water. Cnoidal wave theory is based on equations developed by Korteweg and de Vries The resulting equations contain Jacobian elliptical functions, com- monly designated cn, so the name cnoidal is used to designate this wave theory.
The most commonly used versions of this theory are to the Wrst order, but these theories are still capable of describing Wnite-amplitude waves. The deep water limit of cnoidal theory is the small-amplitude wave theory and the shallow water limit is the solitary wave theory. Owing to the extreme complexity of applying the cnoidal theory, most authors recommend extending the use of the small- amplitude, Stokes higher order, and solitary wave theories to cover as much as possible of the range where cnoidal theory is applicable.
The most commonly used presentation of the cnoidal wave theory is from Wiegel , who synthesized the work of earlier writers and presented results in as practical a form as possible. Elements of this material, including slight modiWcations presented by the U. Army Coastal Engineering Research Center , are presented herein. The reader should consult Wiegel , for more detail and the information necessary to make more extensive cnoidal wave calculations. Some of the basic wave characteristics from cnoidal theory, such as the surface proWle and the wave celerity, can be presented by diagrams that are based on two parameters, namely k2 and Ur.
The parameter k2 is a function of the water depth, the wave length, and the vertical distance up from the bottom to the water surface at the wave crest and trough. It varies from 0 for the small-amplitude limit to 1.
Figures 3. From Figure 3. The Ursell number indicates how appropriate cnoidal theory is for our application and allows the wave length to be calculated if the wave height and water depth are known. Figure 3. Note that when k2 is near zero the surface proWle is nearly sinusoidal, whereas when k2 is close to unity the proWle has a very steep crest and a very Xat trough with the ratio of crest amplitude to wave height approaching unity.
Solution for basic parameters of cnoidal wave theory. ModiWed from Wiegel, Cnoidal wave theory surface proWles. Using cnoidal wave theory determine the wave length and celerity and com- pare the results to the small-amplitude theory. Also plot the wave surface proWle. Solution: To employ Figure 3. Using the procedure demonstrated in Example 2. With the value of k2 and the wave length and height, the surface proWle can be determined from Figure 3.
For cnoidal theory to the Wrst order the pressure distribution is essentially hydrostatic with distance below the water surface, i. Surface proWle and particle paths for a solitary wave. The water particles move forward as depicted in Figure 3.
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