La personalidad y los métodos tendientes a anular la personalidad

Water waves are affected by changes in the bed level, and interact with coastal structures. De- spite the assumptions made in the formulation of linear wave theory, it can be used to explain many of these transformations. Dean and Dalrymple [1991] provide a thorough explanation of all the transformations described below.

As waves move into shallower water, according to equation (2.10) they slow down. For a steady state, the rate of change at which energy is transferred by the waves - the energy flux - remains constant. Therefore, if a wave travels from deep water to shallower depths:

Ecg0=Ecg (2.54)

where E is the wave energy, cgis the group celerity, or the velocity at which wave energy is transmitted, and cg0is the deepwater group celerity. Because the wave energy is proportional

to H2: H H0 =s cg0 cg =Ks (2.55)

which results in an increase in wave height - or wave shoaling - as waves move towards the shore. Ks is the shoaling coefficient; in transitional water depths Ks is less than one, but in- creases above one in shallow water.

Additionally, waves travelling obliquely into varying water depths are subject to wave re- fraction. Wave refraction can be explained by the fact that the number of waves leaving an area per unit time must be the same as the number of waves entering (principle of conservation of waves). For straight and parallel bed contours, this leads to Snell’s law:

sin θ sin θ0

= c

c0 (2.56)

Therefore, the wave fronts bend towards the shoreline as the water depth decreases. In addi- tion, because energy flux is conserved, the equation for shoaling is modified, such that:

H H0 = s cg0 cg r cos θ0 cos θ = KsKr (2.57)

where Kris the refraction coefficient, and its value is always less than one. Because the work in this thesis will consider waves travelling perpendicular to the shore, wave refraction will not be considered. Additionally, wave diffraction will not be considered; this is the phenomenon whereby waves propagate into the shadow of an obstacle such as a breakwater. Wave diffrac- tion is important in the design of ports and harbours; once again the interested reader is re- ferred to Dean and Dalrymple [1991].

Wave reflection is important when considering the interaction of waves with structures, whether or not they are submerged. By considering a wave propagating towards a seawall, the combined incident and reflected wave will be:

ζi+ζr =aicos(kx−ωt) +arcos(kx+ωt) (2.58)

If it is assumed there is no energy loss (i.e. ai=ar =a), then the incident and reflected waves will combine to form a standing wave:

ζ=2a cos kx cos ωt (2.59)

in phase as the wave interacts with and reflects from a structure. Various techniques have been developed to separate the incident and reflected waves in a wave record; the method of Frigaard and Brorsen [1995] will be discussed in Section 5.3.1.

An important concept, with its origins in linear wave theory, is that of radiation stress (alter- natively known as momentum flux). The theory and its application was developed by Longuet- Higgins and Stewart [1964] and describes many nonlinear phenomena associated with waves. The radiation stress is defined by Longuet-Higgins and Stewart [1964] as, “the excess flow of momentum due to the presence of waves”, which can be expressed as (in the x direction):

Sxx = ˆ _ζ −dp +ρu2dz− ˆ ₀ −dp0dz = E  2kd sinh 2kd+ 1 2  (2.60)

There is also a radiation stress in the y direction, Syy:

Syy=E  2kd sinh 2kd  (2.61)

which reduces to zero in deep water. Radiation stress can be used to explain wave setup, which is an increase in the mean water level in the surf zone. As waves break and lose energy, the loss of radiation stress is balanced by an increase in mean water level; this is the phenomenon of wave setup. There is also a lowering of mean water level in the location of shoaling waves seaward of the surf zone (wave set-down), which is also explained by the concept of radiation stress.

When the concept of radiation stress is applied to groups of waves, low frequency infra- gravity waves result (at the shoreline they are known as surf beat). Under higher waves, where the radiation stress is larger, the mean water level is lower than in regions of lower wave energy. This low frequency wave is bound to wave groups travelling at the group celerity; however in the surf zone Longuet-Higgins and Stewart [1964] argued that these waves are released as the higher frequency waves break, and are reflected offshore. However, there is another mechanism for the generation of surf beat; Symonds et al. [1982] showed that these long waves can be generated by the variation in break-point over time. Schäffer et al. [1993] showed that wave groups can still exist inside the surf zone, and that these can contribute to the low frequency waves, alongside the varying break point. An analysis of laboratory data by Baldock et al. [2000] suggested that the varying break-point generation was the predominant cause of surf beat, although Battjes et al. [2004] concluded that there is a strong dependence on the bed slope, with the low frequency bound waves making a greater contribution for milder

slopes.