Splash Zone

Linear wave theory presented so far describes (formally) the water motion in the zone below the still water level. Indeed, as far as the theory is (strictly) concerned, the wave surface never departs from that level z = 0; this comes from the linearizations used. Such a limitation is especially inconvenient when splash zone hydromechanics becomes relatively more important. This can be the case when considering nonlinear phenomena in ship motions as in chapter 9 or when predicting survival loads on o¤shore tower structures in chapter 13. Another di¢culty is that one usually …nds that the wave crest elevation is generally higher than its corresponding trough is deep; the wave is actually somewhat asymmetrical, it does not have a perfect sinus-shape. Higher order wave theories were introduced to solve this latter problem.

Higher Order Wave Theories

In the past, there have been several attempts to alleviate the above limitation of linear theory in order - at least - to better describe the actual wave surface pro…le in mathematical terms, and - with luck - to describe the water kinematics up to the actual wave surface as well.

The theories one can …nd in the - primarily older - literature include the cynoidal theory and Stokes’ second, third and …fth order theories. All of these theories describe only a regular wave, by the way. They are all nonlinear which means that superposition - as described in the beginning of this chapter - cannot be used with them.

Generally speaking the increasing need for statistical information on the sea and a struc-ture’s reaction to it, has led to the more widespread acceptance of linear methods in general and linear wave theory in particular. None of the above mentioned theories will be dis-cussed here, therefore; many of them can still be found in other textbooks, however. Other ways to calculate the water motions in the splash zone have been found.

Pro…le Extension Methods

The …rst step is to realize that - especially for higher waves often used in design - the wave crest will be higher than the sinusoidal wave amplitude above the sea surface. The higher order wave theories, above, included this more or less automatically. When linear theory is used this must be done ’by hand’. It is usually assumed that the wave will extend 0.6 to

0.7 times its height above the still water level; two thirds or six tenths are commonly used fractions for this.

Once this has been taken care of, the next step is to adapt linear wave theory in some e¤ective way. Remember that we can use the following equation to describe the horizontal water velocity component:

u_a(z) = H

2 ¢ ! ¢ cosh k (h + z)

sinh kh (5.95)

This equation is derived from equation 5.57 by substituting ³a= H=2, for use in the range 0= z = ¡h; it is valid for all water depths.

There are some simple pro…le extension methods available:

² Extrapolation

This uses unaltered linear wave theory all the way from the sea bed to the actual water surface elevation or wave crest height; z is simply allowed to be positive in the formulas. Straightforward mathematical extension of linear theory in this way leads to an ’explosion’ in the exponential functions so that the velocities become exaggerated within the wave crest; such results are generally considered to be too big or over-conservative.

² Constant Extension

A second relatively simple method uses conventional linear wave theory below the mean sea level. Then the water velocity found at z = 0 is simply used for all values of z > 0: This is commonly used and is quite simple in a hand calculation.

When the wave pro…le is below the still water level, then one simply works with linear theory up to that level, only.

² Wheeler Pro…le Stretching

Pro…le stretching is a means to make the negative z-axis extend from the actual instantaneous water surface elevation to the sea bed. Many investigators have sug-gested mathematical ways of doing this, but Wheeler’s method presented here is the most commonly accepted, see [Wheeler, 1970]. He literally stretched the pro…le by replacing z (on the right hand side of equation 5.95 only!) by:

z⁰= qz + h(q ¡ 1) in which: q = h

h + & (5.96) where:

z⁰ = a computational vertical coordinate (m): ¡ h 5 z⁰5 0 z = the actual vertical coordinate (m): ¡ h 5 z 5 &

q = a dimensionless ratio (-)

& = the elevation of the actual water surface (m), measured along the z-axis h = the water depth to the still water level (m)

What one is doing, is really computing the water motion at elevation z⁰but using that water motion as if it were at elevation z. Wheeler stretching is a bit cumbersome for use in a purely hand calculation. It can be implemented quite easily in a spreadsheet or other computer program, however. It is popular in practice, too.

One should note the following about the formulas in this section:

² If only a maximum velocity is needed, the time function can be neglected as has been done here. The time function plays no distinctive role in any of these methods.

² All z⁰-values used will be negative (or zero); the exponential functions ’behave them-selves’.

² Each of these approaches degenerate to linear theory if it is applied at the still water level.

² The above formulations are universal in that they can be used in any water depth, at any point under a wave pro…le and at any particular phase within the wave (if desired).

² Since all of these stretching methods are based upon linear wave theory, they can all be used with irregular waves.

Users should be aware that use of deep water wave theory - with its simpli…cations - will lead to less than conservative results as the water depth decreases; this is the reason that the full theory is used above. Indeed, water motion amplitudes within a wave are never smaller than those predicted by deep water theory.

Comparison

A check out of the behavior of all of the above computational procedures in a typical situation is done here for the following condition: H = 15 meters, T = 12 seconds and h = 40 meters. These values (with a crest height which is 2=3 ¢ H) yield q = 40=50 = 0:80:

The computed results are shown in …gure 5.21.

Figure 5.21: Comparison of Computed Horizontal Water Velocities The following observations can be made:

² A consistently extrapolated Airy theory yields the largest values. This might well be expected.

² Similarly, a ’plain’ Airy theory which simply neglects the wave crest entirely, yields a lower bound value.

² All methods presented give identical results at the sea bed - as they should.

² Wheeler stretching seems to yield attractive results in that they lie within the ex-pected range and they are considerably less than the upper bound.